L  B 


UC-NRLF 


I 


BIENNIAL   REPORT 


President  of  the 


OF    CALIFORNIA 


1893 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA 


GIFT  OK 


Accession          90259         Class 


Spiral   Plan 
Teaching  Arithmetic 

BY 
JOHN  H.  WALSH 


I.  THE  SPIRAL  METHOD. 


II.  WHAT  IS  NEEDED  IN  AN  IDEAL 
TEXT  BOOK. 


III.  PLAN  AND  SCOPE  OF  THE  WALSH 
BOOKS. 


130- 


I. 

THE  SPIRAL  METHOD, 


A  brief  description  of  one  or  more   typical   text-  The  Old-time 
books  of  the  old  style  will  enable  the  reader  to  realize  Te3rt-book- 
the  fundamental  defects  of  those  made  after  that  plan. 

The  author  of  a  book  of  that  class  fails  to  under- 
stand that  the  logical  development  of  the  subject  mat- 
ter produces  almost  invariably  a  faulty  arrangement 
of  topics  to  be  followed  by  the  young  pupil.  The  re- 
sult of  his  labors  may  be  a  good  book  of  reference,  but 
a  bad  teaching  book. 

In  the  old  grammars,  the  first  place  was  given  to  Grammar, 
an  explanation  of  the  word  '  'grammar, "  then  one 
of  "English  grammar."  This  was  followed  by  the 
enumeration  of  the  four  parts,  and  a  definition  of  each. 
Next  in  order  would  come  the  elaboration  of  the  first 
part,  covering  many 'pages.  The  very  young  pupil 
was  expected  to  learn  that  "orthography  treats  of  let- 
ters, syllables,  separate  words,  and  spelling;"  after 
which  the  classification  of  the  letters  was  given. 
Vowels  and  consonants  (with  the  double-barreled  w  and 
y),  mutes  and  liquids,  labials,  dentals,  etc.,  would  help 
to  befog  the  youthful  intellect. 

However,  the  student's  troubles  were  only  begin- 
ning. Although  he  might  know  the  declensions,  com- 
parisons and  conjugations  almost  as  well  as  the 


THE  SPIRAL   METHOD. 


Spelling- 
book. 


teacher,  he  didn't  know  that  he  knew  them,  nor  was 
he  taught  how  to  study  them.  Blindly  he  struggled 
for  many  weary  months,  and  with  what  result?  Pos- 
sibly he  became  able  to  repeat  many  meaningless 
forms. 

Geography.  A   geography   text-book   almost  invariably  began 

with  a  definition  of  ''geography,"  then  stated  the 
division  of  the  subject  into  mathematical,  physical, 
political,  etc. ,  with  the  definition  of  each.  Although 
mathematical  geography  is  the  least  interesting,  as 
well  as  the  most  difficult,  for  the  beginner,  it  had  to  be 
completed  before  the  next  division  was  touched;  and 
so  on  with  the  remaining  divisions. 

The  arrangement  of  the  speller  was  frequently 
according  to  the  number  of  syllables  in  the  word — 
monosyllables  taking  the  first  place,  followed  by 
disyllables,  trisyllables,  etc.,  utterly  regardless  of  the 
needs  of  the  pupil.  If  homonyms  were  included,  they 
came  late  in  the  book  and  were  arranged  alphabetically. 
The  child  could  not  learn  the  difference  between  tot 
two,  and  too,  until  he  had  worried  through  the  un- 
familiar words  under  the  earlier  letters — cygnet  and 
signet,  crewel  and  cruel,  fane  and  feign,  for  instance. 

Arithmetic.  In   arithmetic,    after    definitions  of  quantity,  unit, 

number,  concrete,  abstract,  etc.,  the  subject  of  numera- 
tion and  notation  was  reached.  When  the  author 
deemed  this  sufficiently  exhausted — some  books  go  up 
to  twenty  periods,  vigintillions — he  took  up  addition, 
not  forgetting,  however,  to  give  several  principles  and 
rules  in  the  first  topic.  Addition  had  its  definitions, 
principles,  and  rules,  and  a  more  or  less  exhaustive 
treatment.  Then  came  subtraction,  multiplication, 
and  division,  each  containing  its  set  of  definitions> 


THE  SPIRAL   METHOD.  5 

principles,  and  rules,  and  each  being  practically  com- 
pleted before  the  next  was  begun.  This  work  gener- 
ally filled  up  four  years,  all  through  which  many  un- 
fortunates were  compelled  for  various  reasons  to  leave 
school. 

Before  the  sacred  ground  of  fractions  could  be  trod- 
den, a  probationary  period  had  to  be  spent  among  the 
"properties  of  numbers,"  which  gave  the  author  an 
opportunity  to  tax  the  child  mind  with  a  lot  of  things 
about  prime  numbers  and  composite  numbers,  and 
composite  numbers  prime  to  each  other;  with  factors 
and  multiples,  and  common  factors;  with  common 
divisors  and  greatest  common  divisors;  with  common 
multiples  and  least  common  multiples.  Each  new 
name  had  its  definition,  which  made  it  no  clearer;  each 
subdivision  had  its  principles  and  rules;  each  had  its 
set  of  examples — with  possibly  a  few  heart-breaking 
"problems. 

The  subject  of  fractions  afforded  rare  ground  for 
the  old-time  author  and  his  modern  follower.  Defini- 
tions were  given  of fraction  t  fractional  unit ',  unit  of  the 
fraction ,  numerator ',  denominator,  common  fraction,  pro- 
per fraction,  improper  fraction,  simple  fraction,  com- 
pound fraction,  complex  fraction  and  mixed  number. 
Before  the  pupil  was  permitted  to  add  y*  and  ^ ,  he 
had  to  be  taught  how  to  reduce  a  whole  number  to  an 
improper  fraction,  and  a  mixed  number  to  an  improper 
fraction;  a  simple  fraction  to  lowest  terms,  to  higher 
terms,  and  to  given  higher  terms;  fractions  to  equiva- 
lent fractions  with  the  least  common  denominator;  and 
compound  fractions  to  simple  fractions. 

Sufficient  has  been  given  to  show  the  dreariness  of 
the  old  text-book,  especially  in  the  hands  of  a  teacher 


THE  SPIRAL   METHOD 


The  Two- 
book  Series. 


Reform  in 
Text-books. 


who  required  the  memorizing  of  each  definition,  prin- 
ciple and  rule;  although  the  gradual  development  of 
the  subject  in  a  "logical"  order  might  delight  an  aged 
philosopher.  It  is  difficult,  however,  to  defend  a 
scheme  of  instruction  that  would  prevent  a  pupil  from 
seeing  in  a  text-book  how  to  find  the  cost  of  a  half  of 
a  10  cent  pie  until  he  had  been  at  school  nearly  six 
years;  or  the  method  of  calculating  how  much  would 
have  to  be  paid  for  a  pint  of  milk  when  it  sold  for  6 
cents  a  quart,  until  he  had  spent  still  another  year  in 
the  study  of  arithmetic. 

When  the  author  of  this  style  of  text-book  decided 
that  a  two-book  series  was  advisable,  he  did  not  change 
the  "logical"  arrangement.  All  of  the  sins  of  the 
higher  book  generally  appeared  in  the  lower  one,  and 
in  an  aggravated  form  because  of  the  condensation 
necessary  to  make  a  smaller  book.  To  obtain  as  many 
purchasers  as  possible,  the  first  book  of  arithmetic 
contained  nearly  all  the  topics  of  the  higher .  one,  but 
with  many  fewer  examples  for  practice. 

The  German  educators  were  probably  the  first  to 
realize  the  defects  of  the  old-time  text-book,  although 
the  German  pupil  suffered  comparatively  little  from 
its  use,  owing  to  the  slight  dependence  of  his  teacher 
upon  the  book.  When,  however,  it  was  found  that 
the  study  of  grammar  did  not  result  in  any  lessening 
of  the  number  of  mistakes  made  by  a  pupil  in  speaking 
or  in  writing,  the  intelligent  teacher  came  to  under- 
stand that  correctness  in  speaking  and  in  writing  comes 
from  long-continued  practice  in  correct  speaking  and 
writing,  that  many  people  are  correct  in  these  respects, 
who  have  never  studied  technical  grammar,  and  that 
many  others,  able  to  repeat  glibly  all  of  the  rules  of 


THE  SPIRAL   METHOD.  7 

syntax  with  their  exceptions,  habitually  blunder.  It 
is  now  admitted  that  the  science  of  grammar  is  of  no 
use  in  bringing  pupils  to  correct  habits  of  speech,  that 
all  it  can  do  is  to  help  training  in  thought.  In  the 
elementary  school  it  has  no  place  except  in  the  high- 
est classes,  and  there  for  its  disciplinary  rather  than 
for  its  practical  value. 

To  give  the  required  practice  in  correct  talking  and  "Spiral" 
writing,  it  became  necessary  to  develop  a  systematic  Lessons, 
series  of  language  lessons  intended  to  lead  pupils  to 
the  employment  of  the  proper  forms.  In  these  lessons 
the  "spiral"  arrangement  was  necessarily  adopted, 
each  year  getting  its  share  in  drills  in  the  correct  use  of 
the  common  irregular  verbs,  for  instance,  and  in  speak- 
ing and  writing  correctly  the  sentences  likely  to  contain 
mistakes  in  default  of  such  practice.  No  fear  of 
disturbing  the  "logical"  order  of  topics  prevents  the 
maker  of  a  course  of  study  in  language  from  prescrib- 
ing such  lessons  for  pupils  of  even  the  lowest  grades 
as  will  bring  their  ' '  Him  and  me  done  it "  into  something 
more  in  accordance  with  the  best  usage.  He  considers 
the  arrangement  of  the  subject  matter  from  the  stand- 
point of  the  proper  training  of  the  child,  and  leaves 
the  "logical"  arrangement  for  books  of  reference  or 
for  text-books  used  by  students  of  some  maturity. 

The  modern  geography  is  gradually  dropping  its  "Spiral" 
thought-depressing  definitions.  Children  are  led  to 
getting  accurate  notions  of  land  and  water  forms  with- 
out being  compelled  to  memorize  set  collections  of 
words,  to  them  meaningless.  The  subject  is  led  up 
to,  before  the  text-book  is  reached,  in  systematic  oral 
lessons  through  the  lower  classes.  A  new  elementary 
geography  goes  over  the  whole  ground  three  times  in 


8 


THE   SPIRAI,  METHOD. 


"Spiral" 
Spelling. 


"Spiral" 
Arithmetic. 


the  one  book,  adding  a  few  more  details  in  the  second 
treatment,  and  still  more  in  the  third. 

The  winner  of  the  prize  at  an  old  time  spelling- 
match  was  frequently  unable  to  write  a  short  letter 
without  making  some  orthographical  blunders.  The 
new  graded  speller,  besides  providing  for  much  prac- 
tice in  writing  words  correctly,  has  so  changed  the  old 
arrangement  as  to  teach  children  to  use  "their"  or 
"there"  properly  before  the  letter  /  is  reached  in  the 
homonym  subdivision.  Each  series  of  lessons  now 
has  its  proper  share  of  the  things  that  should  be  taught 
a  child  likely  to  leave  school  before  the  middle  of  the 
book  is  studied. 

Text-books  in  arithmetic  have  been  -somewhat 
slow  in  responding  to  the  demand  for  modern  improve- 
ments. Owing  to  the  fact  that  books  are  not  placed 
in  the  hands  of  pupils  of  graded  schools  until  they  are 
nearly  half  through  the  elementary  course,  superin- 
tendents were  able  to  effect  many  changes  for  the 
better  in  the  lower  classes.  Pupils  of  these  grades 
have  been  required  during  the  first  school  year  to  solve 
oral  problems  involving  any  of  the  four  fundamental 
operations,  and  even  to  find  fractional  parts  of  small 
numbers.  The  more  commonly  used  denominate  units 
were  brought  into  the  work  of  these  grades,  and  many 
other  valuable  reforms  were  made.  The  authors,  in 
self-defense,  were  compelled  to  re- write  their  first 
books  ;  but  the  majority  of  them  have  left  in  the  sec- 
ond books  all  of  the  old,  old  faults. 

The  result  has  been  to  handicap  the  superinten- 
dents of  schools  in  a  great  measure.  A  good  course 
in  the  work  of  the  first  four  years,  with  the  details 
carefully  elaborated,  is  followed  in  the  fifth  year  with 


THE   SPIRAL  METHOD.  9 

the  requirement  that  Blank's  Arithmetic  be  studied 
from  page  —  to  page  — .  A  similar  one  is  made  for 
the  remaining  years. 

It  is  time  for  children  to  begin  to  use  books,  if  they 
have  not  had  them  before,  and  it  is  inadvisable  to  sug- 
gest too  much  flitting  about  from  one  part  of  the  book 
to  another.  In  this  way,  pupils  that  have  multiplied 
by  a  mixed  number  in  the  lower  grades,  and  worked 
simple  problems  involving  pounds  and  ounces,  pecks 
and  bushels,  are  compelled  to  drop  these  topics  en- 
tirely until  the  "logical"  order  brings  them  again  into 
view.  The  pupils  are  not  even  permitted,  in  this  re- 
spect, to  "mark  time;"  they  must  retreat,  through  the 
failure  of  the  books  to  furnish  ammunition. 

As  an  illustration  of  the  '  'spiral"  method  in  history,  "Spiral" 
a  plan  followed  in  good  schools  may  be  shown  in  a  few  Teaching1 
words.  Before  the  regular  text-book  is  reached,  the  History, 
subject  is  taught  orally,  being  commenced  in  the  lower 
grades  with  stories  about  historical  personages.  The 
idea  is  to  cover  the  whole  period  of  American  history 
as  frequently  as  possible.  The  first  year's  pupils  are 
told  about,  say,  Columbus,  John  Smith,  Washington, 
Lincoln,  and  McKinley.  The  next  class  is  told  about 
or  reads  about  the  same  persons  with  additional 
details,  and  other  characters  are  introduced ,  say, 
De  Soto,  Penn,  Putnam,  Grant,  Dewey.  In  a  city 
system  in  which  free  books  are  supplied,  a  short  bio- 
graphical history  is  taken  up  and  read  in  a  year.  The 
next  year  another  is  read  and  discussed,  all  this  being 
done  before  history  is  taken  up  as  a  formal  study. 

The  foregoing  method,  while  offensive  to  the  person 
who  would  prefer  to  stick  to  one  book  divided  chron- 
ologically into  as  many  periods  as  there  were  classes 


10  THE  SPIRAL  METHOD. 

"Concentric     studying  history,  is  probably  traceable  to  the  French 
Courses"  in         ,         r  „  Zj  ,, 

France.  plan  of     concentric  courses. 

After  their  defeat  by  the  Prussians,  which  the 
French  attributed  largely  to  the  superior  education  of 
their  opponents,  they  resolved  to  spare  neither  money 
nor  effort  to  increase  the  efficiency  of  their  schools  of 
all  kinds  to  the  greatest  possible  extent.  They  have 
expended  enormous  sums  of  money  in  equipping  the 
schools  and  in  obtaining  the  best  obtainable  talent  to 
teach  and  to  direct  the  teaching.  Under  these  cir- 
cumstances, it  is  not  strange  that  the  French  schools 
during  the  past  twenty-five  years  have  forged  ahead 
more  rapidly  than  those  of  any  other  country  of  the 
world. 

The  striking  feature  of  the  French  system  is  the 
organization  of  the  studies  of  the  elementary  schools 
into  three  "concentric  courses"  of  two  years  each,  the 
pupils  of  these  six  years  corresponding  approximately 
to  our  classes  from  the  third  year  to  the  eighth,  inclu- 
sive. The  first  two  years  are  spent  in  the  '  'maternal 
school,"  in  which  the  teaching  is  chiefly  oral;  but 
during  the  other  six  years  text- books  in  reading, 
language,  geography,  history  and  arithmetic  are  used. 
In  each  of  the  last  four  subjects  a  different  book  is 
taken  up  every  second  year,  each  book  covering  the 
whole  subject,  but  in  a  method  adapted  to  the  capacity 
of  the  student.  The  *  'spiral' '  arrangement  in  France 
is  more  properly  called  the  '  'concentric  circle' '  method. 


II. 

WHAT  IS  NEEDED  IN  AN  IDEAL  TEXT-BOOK. 


To  determine  the  requirements  of  a  good  teaching 
book,  or  series  of  books,  for  elementary  schools,  two 
things  must  be  determined.  One  has  been  alluded  to 
previously — the  proper  arrangement  of  the  subject  mat- 
ter from  the  standpoint  of  the  learner,  which  is  almost 
the  opposite  of  the  "logical"  arrangement  desirable  in 
a  book  of  reference. 

The  other  is    the    careful    consideration    of    the 
arrangement  that  will  best  take  care  of  the  proper  de-  Matter, 
velopment  and  training  of  the  too  large  number  of  un- 
fortunate children  driven  from  school  at  all  stages  of  • 
the    course    through   poverty   or    other    misfortune. 
Two  thirds  of  the  school  children  are  found  in  the 
classes  of  the  first  four  years,  and  nearly  one -third  are  Lif°  of  j™ 
in  the  classes  of  the  second  four  years.     The  number  Pupils, 
in  the  high  schools  constitute  fewer  than  one-fiftieth, 
while  the  number  in  college  constitutes  but  an   inap- 
preciable fraction. 

The  following  figures  taken  from  a  recent  report  of 
a  city  school  system  show  the  number  per  thousand 
of  pupils  on  register  in  classes  of  each  school  year 
from  the  first  to  the  twelfth.  As  tuition,  text-books, 
and  supplies  of  every  kind  are  furnished  free  of  cost, 
the  showing  is  more  favorable  as  to  length  of  school 
life  than  is  likely  to  obtain  in  cities  less  favored  and  in 
the  rural  districts. 

(ii) 


12  AN   IDEAI,  TEXT   BOOK, 

Primary  classes  1st     year  185 

2d         "     173 

3d         "     159 

4th       "     144     661  total 

Grammar  classes         5th       "     125 

6th  ,    "       93 

7th       "       58 

8th       "       35     311      " 

High  school  classes     9th       "       16 

10th       "         7 

llth       "         4 

12th       •'         1 28      " 

1000      " 

While  it  is  impossible  for  the  average  person  to 
draw  accurate  conclusions  from  the  foregoing  figures 
they  show  nevertheless  a  constant  dropping-out  of 
those  that  need  all  our  assistance.  If  a  superinten- 
dent could  be  sure  of  keeping  all  his  pupils  1 2  years 
or  8  years  or  even  4  years,  he  could  so  arrange  his 
course  of  study  as  to  give  a  certain  completeness  to  the 
education  received  up  to  that  time.  The  problem, 
however,  is  somewhat  more  difficult,  especially  as  the 
short  school  life  of  too  many  is  interrupted  by  fre- 
quent absences  from  various  causes. 

The  system  that  provides  for  only  those  that  go  to 
the  high  school  is  doing  a  very  small  share  of  its  pro- 
per work. 

The  ideal  text-book,  therefore,  must  contain  such 
an  arrangement  of  its  subject  matter  as  will  give  some- 
thing substantial  at  as  many  points  in  the  course  as 
possible. 

The  book  that  takes  up  each  topic  as  frequently  as 
possible  helps  out  the  school  life  of  its  user  by  making 


AN   IDEAL  TEXT   BOOK.  13 

it  easier  to  promote  the  boy  or  girl  that  is  a  little 
"  below  grade"  because  of  enforced  absence  or  of  the  Ideal  Book 
possession  of  fewer  brains  than  the  other  pupils.      In  QneTopical 
the  use  of  the  old  time  book,  a  pupil  that  had  failed  to 
master  a  topic  had  no  chance  to  review  it  properly; 
the  user  of  the  ideal  book  should  have  several  op- 
portunities. 

The  maker  of  a  good  text-book  must  not  be  too 
radical.  The  method  given  in  the  previous  chapter,  Selection  of 
of  reading  history  by  covering  the  whole  ground  each  Topics, 
year  or  two,  is  not  applicable,  at  least  in  all  of  its  de- 
tails, in  a  text-book  of  arithmetic.  While  a  boy 
can  easily  understand  all  about  Dewey,  although  the 
latter  belongs  in  the  last  chapter  of  a  chronological 
history,  and  while  he  is  likely  to  be  as  much  interested 
in  the  hero  of  Manila  Bay  as  in  the  Northmen  of  the 
first  chapter,  the  same  is  not  true  of  cube  root  as  com- 
pared with  addition.  The  early  curves  of  the  arithmetic 
"spiral"  should  not  include  too  many  topics,  nor  ones 
too  advanced.  Some  authors,  finding  the  "spiral"  of  some 
method  a  good  one,  have  carried  it  to  too  great  Authors, 
extremes.  Having  convinced  themselves  that  they 
have  discovered  an  ingenious  method  of  simplifying 
an  advanced  topic,  they  work  it  into  an  early  page  of 
their  books.  They  forget,  however,  that  the  impor- 
tant thing  to  do  for  every  pupil  of  the  common  school 
is  to  give  him,  at  the  earliest  possible  moment,  a  work- 
ing familiarity  with  the  fundamental  processes,  the  abil- 
ity to  use  simple  fractions  in  their  commoner  applica- 
tions, and  some  acquaintance  with  the  solution  of  pro- 
blems involving  the  most  commonly  used  denominate 
units.  To  permit  the  child  to  fritter  away  valuable 


14 


AN   IDEAL   TEXT   BOOK. 


Measure- 
ments, Per- 
centage and 
Interest 


Omission  of 
Non-essen- 
tials. 


time  on  less  important  matters,  at  the  risk  of  failing  to 
obtain  the  essentials,  is  an  educational  crime. 

On  the  other  hand,  it  is  unwise  to  allow  the  scholar 
of  the  fifth  and  sixth  years  to  give  his  whole  time  to 
tiresome  drills  in  fractions,  decimals  and  denominate 
numbers  to  the  exclusion  of  even  elementary  lessons 
in  measurements,  percentage,  and  interest.  It  is  possi- 
ble to  give  these  by  the  end  of  the  sixth  year,  or  a  lit- 
tle earlier;  and  no  child  should  be  deprived  of  this 
much  arithmetic,  who  is  forced  through  poverty  to 
give  up  attendance  at  school  before  taking  up  the  work 
of  the  seventh  year.  The  boy  or  girl  able  to  remain 
longer  will  obtain  a  better  knowledge  of  these  topics; 
but  the  others  should  receive  at  least  some  instruction 
therein. 

In  considering  what  should  be  contained  in  the 
ideal  book,  it  must  not  be  forgotten  that  judicious 
omissions  constitute  a  source  of  strength  in  a  teaching 
arithmetic.  There  was  a  time  when  the  author  of  a 
geography  prided  himself  upon  the  multitude  of  details 
that  were  crowded  into  his  maps;  to-day  he  calls 
attention  to  their  small  number.  Arithmetic  is  studied 
to  develop  mathematical  power  in  the  learner,  and  not 
to  give  him  a  mass  of  isolated  facts;  and  the  more  the 
pupil's  attention  is  distracted  by  the  latter,  the  less 
likelihood  there  is  that  he  will  obtain  the  full  benefit 
to  be  derived  from  the  study.  A  boy  that  is  "good  at 
figures"  can  readily  adapt  himself  to  the  arithmetical 
requirements  of  almost  any  calling,  as  soon  as  he 
learns  the  few  facts  peculiar  to  his  position. 

Ability  to  work  examples  involving  denominate 
numbers  can  be  very  much  better  obtained  from  the 
use  of  a  few  tables  containing  familiar  units,  than  from 


AN   IDEAL  TEXT  BOOK.  15 

the  introduction  of  the  other  tables.  The  absence  of 
the  "related  predicates"  in  the  case  of  the  latter,  tends 
very  much  to  the  confusion  of  the  youthful  learner. 
Meeting  a  new  table  a  few  years  later  in  a  new  business 
gives  him  no  trouble,  because  the  daily  routine  fur- 
nishes the  "related  predicates"  that  are  not  present  in 
his  school  days. 

Many  books  prevent  the  pupil  from  testing  his  Unnecessary 
powers.  Each  topic  or  subdivision  of  a  topic  is  treated 
as  if  it  were  something  entirely  new;  and  explanations, 
principles  and  rules  are  furnished  where  their  intro- 
duction is  a  positive  injury  to  the  learner's  develop- 
ment. The  older  authors  were  content  to  give  only 
four  sets  of  rules,  after  they  had  reached  addition  of 
.fractions;  some  newer  ones  work  in  at  least  two  more: 
"To  find  what  fractional  part  one  number  is  of  an- 
other," and  "To  find  the  whole  when  the  fractional 
part  is  given." 

Probably  the  worst  teaching  done  in  our  schools 
occurs  in  the  arithmetic  classes  in  the  seventh  year. 
At  this  stage,  percentage  is  generally  reached;  and  at 
a  time  when  some  mathematical  power  should  have 
been  gained,  the  author  and  teacher  endeavor  to  pre-  Bad  Teach- 
vent  its  display.  The  only  new  thing  in  percentage 
that  the  thirteen-year-old  pupil  needs  to  know  before 
being  set  to  work  at  the  exercises,  is  the  mean- 
ing of  the  term  "per  cent."  Being  told  that,  he 
should  be  able  to  solve  every  question  that  does  not 
contain  any  strange  technical  words.  The  pupil  has 
solved  similar  problems  in  the  fifth  year  in  fractions, 
and  in  the  sixth  year  in  decimals;  and  it  would  seem 
a  pedagogical  blunder  to  force  unnecessary  assistance 


16 


AN   IDEAL  TEXT  BOOK. 


Unnecessary 
Subdivisions. 


New-fangled 
Systems  of 
Arithmetic. 


upon  him,  were  it  not  an  offence  committed  by  authors 
of  high  repute. 

Not  content  with  stifling  all  growth  at  the  outset  of 
the  new  topic  by  their  wrong  treatment  of  it,  these 
authors  present  the  same  thing  again  and  again  under 
such  new  names  as  insurance,  commission,  brokerage, 
profit  and  loss,  taxes,  duties,  and  apparently  endeavor 
to  prevent  a  scholar  from  ascertaining  that  he  is  not 
taking  up  something  strange  by  giving  him  a  set  of 
rules,  principles,  and  cases,  with  each  subdivision.  A 
boy  or  girl  would  be  positively  benefited  by  the  omis- 
sion from  the  text  book  of  every  one  of  these  sub- 
topics. This  would  not  prevent  a  teacher  from  giving 
problems  usually  placed  under  one  or  other  subdivis- 
ions if  she  were  careful  to  avoid  the  introduction  of 
unfamiliar  words,  whose  meaning  could  not  readily  be 
determined  from  the  context.  Any  other  examples 
are  unnecessary.  The  well-taught  pupil  can  handle 
any  he  meets  in  his  particular  business. 

Because  of  the  meagerness  of  the  results  obtained 
from  the  study  of  arithmetic,  some  well-meaning  peda- 
gogues have  assumed  that  the  present  methods  are 
wrong,  root  and  branch.  Instead  of  endeavoring  to 
improve  the  present  system  by  the  needed  reforms, 
they  begin  with  the  assumption  that  nothing  is  right. 
They  wish  to  kill  the  rats  by  burning  the  barn,  a 
rather  wasteful  procedure.  The  Grube  method  was 
the  first,  in  recent  years,  to  obtain  any  wide-spread  ac- 
ceptance. The  good  things  in  this  method  are  still 
employed  in  a  modified  way;  but  the  interminable 
grind  prescribed  by  its  author  is  not  now  carried  out 
by  any  sane  teacher. 

Two  new  methods,  each  guaranteed  to  be  a  specific 


AN   IDEAL  TEXT   BOOK.  17 

for  all  the  mathematical  ills  we  suffer,  have  recently  The  Rational 
been  proposed  in  all  seriousness.  The  first  assumes  Method- 
that  failure  to  teach  arithmetic  properly  is  due  to  the 
adherence  of  teachers  to  the  "fixed  unit"  of  the  Grube 
system.  If  the  current  practice  of  the  schools  were  to 
begin  work  in  number  with  an  elaborate  drill  on  the 
number  one,  there  might  be  some  reason  for  writing  an 
article  to  show  the  absurdity  of  such  procedure;  but  as 
no  teacher  does  anything  of  the  kind,  the  author  of 
the  "movable  unit"  method  is  threshing  the  air.  If 
teachers  were  bound  to  inflict  upon  babes  tiresome 
drills  on  the  "unit,"  it  would  be  well  to  suggest  the 
employment  of  the  "movable"  one;  but  as  teachers,  in 
the  main,  are  rational  beings,  they  will  not  worry 
about  either  variety  of  unit.  How  ridiculous  it  is  to 
waste  energy  in  teaching  children  what  they  know 
already;  or  what  they  will  inevitably  learn  without 
teaching,  and  learn  better.  What  need  to  drill  elabo- 
rately on  the  facts  that  1  cent  is  different  from  1  nickel, 
that  1  hour  is  not  1  day,  etc.,  etc.  The  only  new 
things  introduced  into  the  teaching  of  arithmetic  by  the 
author  of  the  "Rational  method"  are  the  worse  than 
useless  ones  of  obscuring  the  terms  of  a  straight-for- 
ward problem  by  the  introduction  of  references  to  his 
"movable"  units.  The  text-book  written  to  exploit 
the  method  would  be  vastly  improved  by  the  omission 
of  every  reference  thereto.  It  contains  many  good 
things,  but  they  are  all  old;  what  is  new  in  it  is  bad 
(for  teaching  purposes). 

The  zealous  introducer  of   the   "ratio"  method  The  "Ratio' 
points  with  pride  to  the  wonderful  results  produced  by 
the  use  of  his  system,   with  its  elaborate  outfit  of 
splints,  squares,  rectangles,  triangles,  cubes,  prisms, 


18  AN   IDEAL  TEXT  BOOK. 

cones,  and  what  not.  The  same  remarkable  things 
are  claimed  by  each  inventor  of  something  new  in 
education.  Some  of  the  lookers-on  are  deluded  into 
the  belief  that  the  philosopher's  stone  has  been  found 
at  last.  'Others,  more  experienced,  understand  that 
children  can  be  taught  anything;  but  they  realize  that 
the  price  paid  may  be  the  irreparable  injury  of  the 
children.  Bright  teachers,  proud  of  the  advertising 
they  get  from  streams  of  visitors,  and  desirous  of 
obtaining  the  commendation  of  their  superiors,  will 
fail  to  appreciate  the  harm  they  are  doing  their  young 
charges  by  prematurely  forcing  them  into  tasks  beyond 
their  years.  The  complicated  ratios  introduced  by 
this  system  into  the  number  work  of  the  lowest  years, 
will  inevitably  do  much  harm  to  the  development  of 
immature  minds.  What  is  sought  of  the  babe  can  be 
obtained  much  better  if  postponed  until  the  proper  time. 
Too  much  One  thing  that  will  inevitably  militate  against  the 

Machinery,  success  of  a  method  requiring  too  much  machinery  in 
its  operations  is  the  failure  of  the  over-burdened 
teacher  to  employ  the  necessary  material,  even  when 
it  is  supplied,  and  the  unwillingness  of  the  school 
authorities  to  continue  to  furnish  supplies  that  are 
kept  unused  in  dark  closets.  While  teachers  in  city 
schools,  having  only  pupils  of  a  single  grade,  may 
struggle  for  a  time  with  the  method,  it  cannot  possibly 
meet  with  any  favor  at  the  hands  of  teachers  having 
pupils  of  more  than  one  grade.  The  very  best  results 
in  objective  teaching  are  generally  found  in  the  classes 
in  which  the  objective  material  is  obtained  through 
efforts  of  the  teacher  or  pupils.  Home-made  apparatus 
of  nearly  every  kind  is  much  more  effective  in  class 
work  than  that  supplied  by  school  boards. 


AN  IDEAL  TEXT  BOOK.  19 

It  is  not  so  much  a  new  method  of  teaching  arith-  Science  of 
metic  that  is  needed,  as  a  modification  of  existing  ^jJrtoT 
methods.  In  the  first  place,  teachers  must  understand  Computing, 
that  babes  have  no  business  with  the  science  of  number; 
the  main  object  during  the  early  school  years  should 
be  to  teach  the  art  of  computing.  This  makes  un- 
necessary all  definitions  and  principles.  When  notation 
and  numeration  are  taught  beginners,  there  should  be 
no  wasteful  elaboration  of  method  of  showing  the  value 
of  the  figure  in  the  tens'  place  or  of  the  one  in  the 
hundreds'  place,  the  bundles  of  ten  splints  and  the 
larger  ones  of  a  hundred  splints.  Young  children  can 
learn  to  read  and  write  39  without  much  trouble;  all 
the  time  spent  in  endeavoring  to  give  them  an  adequate 
notion  of  39  is  thrown  away.  The  little  parrots  can 
be  made  to  say  anything  the  teacher  wishes  them  to 
say;  they  can  learn  to  manipulate  the  single  splints  and 
the  bundles  of  ten;  but  the  speeches  and  the  per- 
formance are  meaningless  to  the  baby  actors,  although 
their  teacher  does  not  always  realize  it. 

When  written  addition  is  to  be  taught,  the  teacher  The  "how": 
should  show  his  pupils  the  "how"  of  the  carrying,  not 
the  "why."  This  is  the  opinion  of  all  the  soundest 
educators  of  our  country,  even  if  it  disagrees  with  the 
practice  of  the  girl  whose  normal-school  diploma  is  a 
few  months  old.  While  she  may  think  the  children 
know  what  they  are  talking  about,  when  they  glibly 
rattle  off  the  unmeaning  formulas  prescribed  by  her, 
she  is  seriously  deluding  herself.  It  is  not  claimed,  of 
course,  that  children  are  much  injured  by  this  simple 
teacher's  methods;  a  kind  Providence  has  ordained 
that  the  infantile  mind  does  not  waste  much  effort  in~ 
the  endeavor  to  grasp  matters  entirely  beyond  its 


20 


AN   IDEAL  TEXT   BOOK. 


Lines  of 
Work. 


Oral 
Problems. 


Drills. 


range.  Still,  the  teacher  is  wasting  time;  and,  when 
she  ascertains  that  her  pupils'  previous  repetitions  of 
things  she  wished  them  to  say,  mean  nothing  to  them, 
she  may  do  harm  by  endeavoring  to  force  them  to 
understand. 

Of  course,  oral  problems  are  to  occupy  a  fair  por- 
tion of  the  time  given  to  number  work  and  even  some 
written  problems  should  be  given;  but  a  large  part  of 
the  energy  of  the  beginning  pupils  should  be  given  to 
abstract  written  addition  and  to  the  necessary  oral 
drills.  The  following  are  the  lines  of  work  that 
should  be  followed  throughout  the  arithmetical  course: 
1st,  oral  problems,  to  develop  proper  ideas  of  number, 
and  to  strengthen  the  reasoning  powers;  2d,  oral  drills 
in  rapidly  combining  abstract  numbers,  to  give  the 
necessary  facility  in  performing  written  operations; 
3d,  written  addition,  subtraction,  etc.,  of  abstract 
numbers;  4th,  written  solution  of  problems. 

The  oral  problems  will  be  of  two  kinds:  First, 
those  in  which  the  child  will  have  to  determine  for 
himself  the  operation  involved;  and  these  should  con- 
tain small  numbers,  so  that  the  size  of  the  numbers 
will  not  make  the  problem  unnecessarily  difficult, 
while  failing  to  help  the  pupil's  mathematical  develop- 
ment. In  those  of  the  second  kind,  which  hardly 
deserve  to  be  called  "problems,"  except  that  they  are 
"concrete,"  the  chief  object  is  to  drill  the  pupil  in 
making  the  needed  combination  rapidly  without  a 
pencil;  they  should  involve,  therefore,  as  a  rule,  but 
one  operation. 

The  oral  drills  should  be  given  very  frequently,  but 
for  very  short  periods;  and  they  should  demand  rapid 
answers  on  the  part  of  the  pupils.  These  help  the 


AN   IDEAL  TEXT   BOOK.  21 

scholars  in  their  subsequent  abstract   written  work. 
Sight  drills  must  not  be  overlooked. 

The  written  problems  show  the  pupils  the  applica- 
tions of  the  fundamental  processes  to  the  ordinary 
affairs  of  life. 

The  ideal   arithmetic  should   contain  each  of   the  Careful 
foregoing  lines  of  work,  carefully  gradedf  and  in  the  Gradation 
proper  proportion.     The  book  should  be  divided  into 
sections,  each  containing  the  work  of  a  year  or  a  half 
year.     The  attempt  to  divide  by  daily  lessons  is  un- 
wise; as  nobody  but  the  teacher  can  determine  just 
what  subject  requires  special  attention  at  a  given  time, 
with  a  corresponding  number  of  examples. 

The  abstract  work  should  be  developed  very  slowly  Abstract 
and  carefully,  with  a  very,  very  large  number  of  ex- 
amples. Difficulties  should  be  encountered  one  at  a 
time.  As  facility  in  calculation  is  obtained  only  by 
very  much  practice,  it  is  almost  impossible  to  furnish 
too  many  abstract  examples.  The  task  of  selecting 
them  from  other  arithmetics,  or  of  making  them, 
should  not  be  thrown  upon  the  over-burdened  teacher; 
and  it  is  better  for  the  pupils  to  have  them  in  their 
books  than  to  be  compelled  to  copy  them  from  the 
blackboard.  Reviews  of  processes  already  learned 
should  be  continued  to  the  end  of  school  life.  The 
complaints  of  college  professors  and  of  business  men 
that  graduates  of  high  schools  cannot  even  add  or 
multiply  correctly  is  only  too  well  founded. 

As  the  chief  object  of  the  written  problems  is  to  written 
develop  reasoning  power,   they  must,    if  possible,  be  Problems, 
even  more  carefully  graded  than  the  abstract  work. 
A  problem  should  generally  require   no  explanation 
from  the  teacher;  the  conditions,  therefore,  should  be 


22 


AN   IDEAL  TEXT   BOOK. 


"Miscellane- 
ous" 
Problems. 


Analysis  of 
Problems* 


such  as  are  reasonably  familiar  to  the  pupil.  This  le- 
quires  an  author  to  skillfully  adapt  the  wording  of  a 
problem,  and  the  matters  contained  in  it,  to  the  age  of 
the  average  pupil  that  will  be  required  to  work  it.  The 
numbers  employed  should  not  be  so  large  as  to  daze 
the  learner;  since  the  latter,  as  a  rule,  stumbles  over  a 
written  problem  expressed  in  the  same  words  and  con- 
taining the  same  conditions  as  an  oral  one  with  smaller 
figures  that  gives  him  no  trouble  whatever.  This  lat- 
ter difficulty  can  frequently  be  overcome  by  using 
written  problems  as  "sight"  ones,  the  numbers  being 
changed  by  the  pupil  to  those  small  enough  to  be  han- 
dled without  a  pencil,  and  the  pupil  then  working  the 
problem  again  with  the  figures  given  in  the  book. 

Problems,  both  oral  and  written,  should  always  be 
"miscellaneous;"  that  is,  there  should  be  no  heading 
to  indicate  the  character  of  the  operation  required  to 
solve  one.  If  the  scholar  knows  that  all  the  so-called 
problems  on  a  given  page  involve  multiplication,  he 
need  give  no  attention  to  the  conditions.  No  two 
consecutive  problems  should  involve  the  same  opera- 
tions, except  in  the  few  cases  where  two  or  more  suc- 
cessive ones  are  introduced  to  lead  up  to  a  more  com- 
plicated one. 

While  problems  constitute  a  very  important  part  of 
a  child's  arithmetical  training,  progress  in  abstract 
work  should  not  be  retarded  by  the  teacher,  in  the 
vain  attempt  to  make  each  pupil  work  out  and  under - 
,jgtand  every  problem.  The  wise  teacher  will  not  re- 
quire every  member  of  the  class  to  keep  pace  with  his 
mates  in  problems,  nor  will  she  require  all  to  work  the 
same  problems  at  the  same  time,  nor  will  she  insist 
upon  any  pupil  solving  all  of  them.  She  will  also  in 


AN  IDEAL   TEXT   BOOK.  23 

problem  work,  both  oral  and  written,  assume  that  the 
child  has  reasoned  correctly  when  he  gets  the  correct 
answer;  and,  therefore,  she  will  not  exact  an  unmean- 
ing analysis,  oral  or  written.  If  she  wishes  the  pupil 
to  give  a  reason,  she  will  take  any  one  suited  to  the 
child's  years,  even  if  it  does  not  contain  the  conven- 
tional "wherefore"  or  "hence."  She  will  not  expect 
written  solutions  to  follow  a  set  plan,  nor  will  she 
spoil  a^  bright  boy  by  requiring  him  to  place  upon  his 
paper  any  more  figures  than  are  necessary  to  aid  him 
in  obtaining  the  result.  An  occasional  written  analy- 
sis, as  an  exercise  in  composition,  she  will  probably 
ask. 

While  the  science  of  arithmetic  has  no  place  in  the 
elementary  school,  and  while  the  "how"  should  gen-  Of  Power 
erally  precede  the  "why,"  the  thoughtful  instructor 
will  give  her  brighter  pupils  an  opportunity  to  develop 
power  by  reasoning  out  the  "how,"  before  she  shows 
it  to  the  slower  ones.  After  pupils  have  added  num- 
bers of  two  figures  that  need  no  "carrying,"  she 
might  lead  up  to  the  latter,  by  asking  on  the  board 
the  sum  of  12  and  12,  then  13  and  13,  then  14  and  14, 
then  15  and  15.  Some  members  of  her  class  would 
know  that  the  sum  of  the  last  set  is  30,  even  if  the 
oral  combinations  had  been  confined  to  numbers  of  one 
figure;  and  they  would  appreciate  the  method  that 
produces  this  result.  Subtraction  as  the  reverse  of 
addition  should  be  led  up  to;  but  no  time  should 
be  spent  in  explaining  the  reason  for  the  method  em- 
ployed. Those  that  can  get  any  benefit  from  the  per- 
liminary  work,  should  be  permitted  to  get  it;  but  as 
the  "why"  will  come  anyhow,  although  later,  the 
progress  of  the  class  should  not  be  delayed.  The 


24 


AN   IDEAL  TEXT   BOOK. 


Development 
of  "Rules." 


Topical 
Work. 


Arithmetic 
vs.  Algebra. 


addition  of  15  and  15  will  show  pupils  how  to  get  the 
product  of  15  by  2,  after  they  have  worked  multiplica- 
tion examples  in  which  there  is  no  "carrying." 

As  the  child  grows  older,  and  as  his  reasoning 
power  increases,  he  should  be  given  more  frequent 
opportunities  to  develop  for  himself  the  so-called 
"rules."  All  that  is  necessary  in  the  case  of  most 
children  is  that  the  teacher  should  furnish  examples 
in  which  the  successive  steps  occur  in  the  proper  order. 
She  need  scarcely  say  a  word  in  explanation;  in  fact, 
the  better  the  teacher  the  fewer  her  words,  if  she  takes 
care  to  present  difficulties  no  more  rapidly  than  they 
can  be  surmounted. 

The  working-out  of  this  kind  of  work  is  hardly 
possible  in  the  old-time  book.  Bach  topic  has  to  be 
contained  in  a  chapter,  of  which  the  padding  consti- 
tutes a  large  part.  Work  that  is  best  spread  over  a 
year  or  so,  and  in  which  results  are  to  be  obtained 
only  after  working  hundreds  of  examples,  has  to  be 
dismissed  in  a  few  pages. 

The  erroneous  notion  formerly  prevalent,  that 
pupils  were  somehow  improved  by  over-difficult  tasks, 
prevented  the  adoption  in  this  country  of  the  use  of 
algebraic  methods  by  students  of  arithmetic.  While 
the  old-timer  was  ready  to  admit  that  some  of  the  so- 
called  arithmetic  problems  could  be  solved  in  a  quar- 
ter of  the  time  by  the  use  of  the  letter  x,  he  pretended 
to  believe  that  it  was  a  good  thing  for  the  child  to 
make  him  attempt  the  almost  impossible.  Although 
Spartan  methods  of  killing  off  the  weaklings  have  long 
been  discontinued  on  the  physical  side,  the  school  life 
of  many  has  been  made  unnecessarily  severe  by  requir- 
ing them  to  attempt  to  solve  problems  in  the  hardest 


AN  IDEAL  TEXT   BOOK.  25 

way,  because  of  the  supposed  mental  growth  thereby 
induced.  It  signifies  nothing  to  the  believers  in  this 
plan  that  much  the  larger  number  of  pupils  get  nothing 
out  of  it;  they  are  content  at  the  success  of  the  for- 
tunate few. 

The  opponents  of  the  introduction  of  algebraic  Formal 
methods  into  the  elementary  school  claim  that  this 
subject  should  be  confined  exclusively  to  those  able  to 
go  to  a  high  school.  While  it  may  be  true  that  the 
formal  study  of  algebra  as  a  science  should  not  be 
commenced  until  about  the  ninth  school  year,  no  one 
will  have  the  hardihood  to  assert  that  other  pupils 
should  be  deprived  of  the  use  of  anything  that  would 
tend  to  lighten  their  burdens. 

Even  the  fiercest  opponent  of  the  use  of  x-  in  a  so- 
called  arithmetical  problem,  will   himself,    in   solving  Arithmetical 
such  a  problem,  make  use  of  an   algebraic    method;  Circumlocu- 
but  to  conceal  the  latter,  he  will  explain  his  solution  * 
by  a  tedious  circumlocution  in  which   the  tale-telling 
x  is  carefully  kept  from  view. 

There  is  nothing  in  the  application  of  algebraic  The  use  Of  ^ 
methods  to  the  solution  of  problems  that  is  beyond 
the  capacity  of  the  average  child  or  teacher.  If  in 
a  first-year  class-room,  you  put  on  the  blackboard 
3+?= 5,  the  correct  answer  is  given  by  nearly  every 
pupil.  While  no  one  would  object  to  such  a  problem 
as  an  arithmetical  one,  it  is  called  algebra  when  an  x 
is  substituted  for  the  interrogation  point.  Is  this  fair? 

The  mind  of  the  average  child  works  forward,  so  Going 
to  speak,  better  than  it  works  backward.     A  beginner  Forward' 
in  subtraction  can  tell  the  answer  to  "7  and  what  makes 
10?"  more  readily  than  to  ' '  1 0  less  7  equals  what?' '  The 
former  is  algebraic  in  a  way;  the  latter,  arithmetical. 


26  AN  IDEAI,  TEXT  BOOK. 

The  following  problem,  frequently  found  in  the 
arithmetics,  cannot  be  solved  arithmetically  without 
help,  by  any  but  the  most  phenomenal  scholars  of  the 
age  at  which  they  reach  it  in  the  books:  "The  votes 
cast  for  A  and  B  number  6836,  of  which  A  gets  a 
majority  of  748.  How  many  does  each  receive?" 
An  author  (who  has  studied  algebra)  furnishes  a  neat 
explanation.  His  excuse  for  inserting  this  problem  and 
many  other  similar  ones  is  that  children  may  meet 
them  later  in  life.  Instead  of  furnishing  the  general 
(algebraic)  method,  he  gives  a  different,  pretendedly 
arithmetical  one  for  each,  and  the  scholar  can  solve 
only  those  whose  types  he  has  already  had,  instead 
of  being  enabled  by  the  algebraic  master-key  to  open 
any  lock,  whether  he  has  seen  it  before  or  not. 

Problems  in  Let   us   see  how  the  "problems"    in   interest  are 

Interest.  handled  by  arithmetic  makers  that  are  unwilling  to 
give  their  clients  the  benefit  of  the  simple  methods. 
To  find  what  principal  will  produce  $64  in  3yr.  6mo. 
9da.  at  6%,  they  assume  a  principal  of  $1,  on  which 
the  interest  is  calculated  at  the  given  rate  for  the 
given  time.  The  result  is  found  to  be  $0.2115.  The 
pupil  is  then  told  that  the  required  principal  is  as 
many  times  $1  as  $64  is  times  $0.2115. 

While  the  bright  boy  that  had  never  seen  algebra, 
could  not  of  himself  evolve  the  foregoing  method,  the 
bright  student  of  algebra  could  solve  the  problem 
without  having  encountered  it  previously.  Instead  of 
finding  the  interest  on  $1  (which  is  almost  as  much 
algebra  as  arithmetic)  he  would  calculate  it  on  x  dollars, 
and  call  this  result — .2115:*: — equal  to  64,  the  equation 
being,  without  decimals,  2115^=640,000.  His  prac- 
tice in  solving  equations  suggests  the  remaining  step. 


AN  IDEAL  TEXT  BOOK.  27 

These  interest  problems  are  at  least  six  in  number: 

1 .  Given  interest,  rate,  time,  to  find  principal. 

2.  Given  principal,  rate,  interest,  to  find  time. 

3.  Given  principal,  time,  interest,  to  find  rate. 

4.  Given  amount,  rate,  time,  to  find  principal. 

5.  Given  amount,  rate,  principal,  to  find  time. 

6.  Given  amount,  time,  principal,  to  find  rate. 
While  others  might  be  enumerated,  the  foregoing 

are  sufficient  to  show  the  advisability  of  applying 
algebraic  methods  when  they  are  available.  These 
six  "problems"  really  should  constitute  no  "problem" 
of  special  note  from  the  algebraic  standpoint.  Any- 
one that  knows  a  little  of  algebraic  ways,  and  able  to 
calculate  interest,  needs  no  "rule".  He  works  out  the 
interest  (or  amount) ,  using  x  to  represent  the  missing 
factor,  and  he  then  forms  the  necessary  equation.  By  vs. 
following  the  old  way,  the  learner  is  required  to  mem- 
orize  six  rules;  especially  as  the  text-book  maker  fails 
to  make  clear  the  connection  between  them.  Not 
knowing  the  underlying  reasons,  he  forgets  the  rules 
shortly  after  leaving  school,  and  he  becomes  helpless. 
The  algebra  boy  gets  no  "rule"  for  solving  any  prob- 
lem, except  to  treat  the  x  just  as  he  would  the  number 
it  represents. 

As  most  teachers  are  from  high  or  normal  schools,   Easy  to 
or  both,  but  very  few  will  be  compelled   to   make   any 
special  study  of  the  algebraic  method.     These   few,    if 
competent  to  teach  arithmetic,    can   make  themselves 
familiar  with  the  method  after  very  little  practice. 

The  writer  of  these  lines  deprecates  the  attempts  of 
some  misguided  people  who  seek   to  introduce  formal  High  school 
algebra  into  the  grammar  school  course,  reasoning  that  A1gebra- 
it  will  save  time  later  in  the  high  school.         The 


28 


AN 


TEXT   BOOK. 


Constructive 
Work. 


elementary  course  is  intended  primarily  for  those  whose 
education  ends  there.  Nothing  should  be  introduced 
that  needs  high  school  work  to  give  it  any  value.  A 
boy  that  gives  a  year  or  so  to  addition,  subtraction,  etc. 
of  algebraic  quantities  gets  no  benefit  whatever  from 
the  study,  if  he  does  not  go  to  high  school.  This  time 
should  be  given  almost  entirely  to  the  solution  of  the 
equation.  The  pupils  that  go  to  the  high  school  should 
begin  the  study  of  algebra  at  the  beginning,  just  the 
same  as  those  who  had  never  seen  it,  except  that  they 
may  progress  much  more  rapidly  from  the  insight 
given  them  of  the  utility  of  the  new  study.  Diluted 
high  school  algebra  should  have  no  place  in  an  eight 
years'  course  of  study  for  the  elementary  schools. 
Those  who  propose  it  do  not  appreciate  the  real  prov- 
ince of  "algebra  below  the  high  school." 

While  a  few  French  and  German  schools  postpone 
algebra  until  a  late  period  of  the  elementary  school 
course,  they  all  introduce  geometry  very  early  in  the 
child's  school  life,  the  French  schools  beginning  it 
during  the  first  year. 

As  we  do  not  teach  high  school  algebra  when  we  use 
x  in  the  arithmetic  work  of  the  lower  schools,  the  word 
"geometry"  as  employed  in  European  courses  of  study 
does  not  mean  that  demonstrative  geometry  is  inflicted 
upon  infants.  Geometry  is  begun  in  France,  as  in  the 
United  States,  by  clay  modeling,  drawing  squares  and 
circles,  constructing  paper  prisms  and  cones.  It  in- 
cludes mensuration  of  surfaces  and  of  solids,  as  well  as 
the  solution  of  simple  problems  in  construction.  It 
does  not  mean  the  study  of  the  high  school  text-book. 

For  every  pupil  of  the  elementary  school  that  will 
be  called  upon  to  work  out  an  example  in  partial  pay- 


AN   IDEAL  TEXT   BOOK.  29 

ments,  there  will  be  a  thousand  likely  to  require  some 
knowledge  of  geometrical  facts.  These  facts  should 
be  commenced  in  the  lower  school  at  as  early  an  age  as 
possible.  The  drawing  courses  in  many  schools, 
urban  and  rural,  provide  the  best  possible  instruction 
for  the  smaller  children.  By  the  end  of  the  fourth  utility  of 
year  systematic  work  should  come  into  the  arithmetic  GeometlT- 
work,  a  little  at  a  time.  Mensuration  of  rectangles, 
formerly  left  for  the  end  of  the  eighth  year,  and  later 
pushed  forward  a  year  or  so,  should  commence  at  the 
beginning  of  the  fifth  year.  Each  year  of  the  last  four 
should  contain  its  proper  share  of  calculating  surfaces 
and  volumes;  and  the  last  year  or  two  should  contain 
a  well  worked-out  set  of.  construction  problems,  the 
working  of  which  would  put  the  pupil  in  possession  of 
the  most  important  facts  of  geometry. 

As  in  algebra,  the  injudicious  should  not  be  per- 
mitted to  push  demonstrative  geometry  into  elementary 
schools.     The  latter  is  studied  in  the  high  school  for 
its  disciplinary  value.     The  constructive  work  of  the 
other,  besides  being  extremely  valuable  to  those  quit-  Demonstra. 
ting  school  at  the  end  of  the  eighth  year,  is  also  useful  tive  Geom- 
to  the  later  student  of  Euclid  or  of  Legendre.     Know-  etry' 
ing  the  facts,  he  is  better  able  to  appreciate  the  chain 
of  reasoning  employed  in  the  text-book. 

It  is  hardly  necessary  to  enumerate  the  persons  to 
whom  some  knowledge  of  geometrical  facts  is  useful. 
The  farmer,  mason,  plasterer,  carpenter,  tinsmith, 
painter,  have  all  to  deal  with  mensuration  to  a  greater 
or  less  extent,  and  also  with  the  construction  of  some 
of  the  geometrical  forms. 

The  work  in  constructive  geometry  can  be  done  by  Easily 
comparatively     young    children.       What  the    latter  Tau2ht- 


30 


AN  IDEAL  TEXT   BOOK. 


How  to  Find 
Time. 


Cutting  out 

Useless 

Topics. 


can  readily  do,  should  present  no  difficulty  to  the 
teacher. 

The  question  of  time  is  always  an  important  one  in 
the  present  over-crowded  course.  Friends  (or  enemies) 
of  the  schools  are  continually  coming  to  the  front  with 
additional  studies  to  be  inflicted  upon  the  system,  and 
those  teachers  are  hardly  blamable  who  look  with 
disfavor  upon  the  attempt  to  add  algebra  and  geometry 
to  the  over-loaded  curriculum.  If,  however,  they 
realize  that  there  is  no  desire  to  introduce  the  high- 
school  subjects  known  by  these  names,  that  all  that  is 
intended  is  to  suggest  the  employment  of  a  few  simple 
algebraic  expedients  in  solving  arithmetic  problems, 
and  the  use  of  drawing  as  an  aid  in  mensuration  work, 
their  objections  will  be  less  vigorous. 

The  benefits  to  teachers  and  scholars  will  be  in- 
creased if  the  adoption  of  the  new  method  leads  to  bet- 
ter arrangement  of  the  old  topics  and  the  elimination 
of  every  unnecessary  one.  By  cutting  out  the  things 
that  now  overload  the  books,  the  time  at  present  given 
to  arithmetic  may  be  lessened,  the  subject  will  be  bet- 
ter taught,  and  plenty  of  time  will  be  found  to  use  the 
equation  and  to  work  out  problems  in  construction. 


III. 

PLAN  AND  SCOPE  OF  THE  WALSH  BOOKS. 

The  Walsh  arithmetics  constitute  a  one-book  series 
bound  for  convenience  in  two  or  in  three  parts.  The 
first  page  of  one  book  follows  immediately  after  the 
last  page  of  the  preceding  one,  without  a  break.  The 
purchaser  of  the  second  book  does  not  buy  a  number 
of  useless  pages,  as  he  must  frequently  do  in  the 
case  of  other  series. 

Each   arithmetic  chapter   after   the   first  contains  Half-year 
work  for  a  half  year.     Besides  the  appropriate  advance       ap  er8' 
work  in  all  the  lines,  oral  and  written,  drill  work  and 
problems,  it  contains  the  necessary  reviews. 

Children  will  not  willingly  turn  backward  to  get  mat-  Reviews 
ter  for  reviews.  The  advisability  of  taking  new  lessons 
for  this  purpose  is  especially  appreciated  by  teachers  of 
language,  English  and  -foreign,  ancient  and  modern. 
Besides,  the  need  of  constant  review  is  likely  to  be 
overlooked  unless  matter  for  the  purpose  is  brought 
directly  to  the  teacher's  attention. 

Another  strong  feature  is  the  careful  grading.  In  Large  Num- 
the  abstract  work,  the  examples  are  so  numerous  that 
the  difficulties  are  introduced  as  slowly  as  is  compati- 
ble with  good  work.  The  very  great  number  of  the 
abstract  examples  gives  the  needed  facility  in  rapid 
and  accurate  calculation.  The  examples  are  so  graded 
that  the  child  can  begin  written  work  early  in  his 
school  life,  and  continue  it  without  interruption.  He 
can  begin  to  add  as  soon  as  he  knows*  a  few  sums,  and 

(31) 


32  THE   WAI£H   ARITHMETICS. 

the  successive  examples  grow  difficult  by  slow  degrees. 
In  the  first  nine  pages,  for  instance,  there  are  258  ex- 
amples in  addition,  oral  and  written,  with  the  total  of 
each  column  less  than  10.  The  children  learn  to  do 
by  doing.  Then  follow  121  examples,  problems,  etc., 
leading  to  written  subtraction,  without  ''borrowing,' 
then  65  examples  in  written  subtraction  and  10  *  'mis- 
cellaneous" problems,  454  examples  in  all  before  10  is 
used  as  a  sum  of  any  column  or  as  a  separate  minuend. 
In  addition  to  the  foregoing  there  are  93  exercises  in 
numeration  and  notation  of  numbers  to  99,  or  547  ex- 
ercises of  all  kinds  in  the  first  fourteen  pages.  This 
shows  the  absence  of  padding  and  the  care  taken  in 
the  development  of  the  work.  The  remaining  21 
pages  of  the  first  chapter,  devoted  to  addition  and  sub- 
traction of  easy  numbers,  contain  879  exercises  of 
all  kinds,  including  drills,  etc.,  making  a  total  of  1426 
for  35  pages.  These  will  be  none  too  many,  as  the 
work  of  this  chapter  is  intended  to  cover  what  is 
usually  done  by  the  end  of  the  second  year. 

The  next  chapter  is  arranged  for  pupils  of  the  first 

half  of  the  third  school  year.     It  extends  the  previous 
Multiplica- 
tion and          work  in  addition  and  subtraction,  and  takes  up  multi- 
Division,         plication  and  division,  beginning  with  the  new  work. 
Multiplication  is  commenced  at  once  on  the  supposi- 
tion that  the  child  has  learned  from  hi-s  addition  work 
the  products  by  2  up  to  twice  4.     Ten  exercises  and 
60  examples   are  given  with   2    as  a  multiplier,  and 
without  carrying,  to  fasten  the  child's  knowledge  of 
the    early    table.     The    last    10   examples  introduce 
larger  products,  but  the  examples  still  involve  no  car- 
rying.    Division  by  two  is  next  taken  up,  with  each 
figure  of  the  dividend  a  multiple  of  the  divisor,  the 


THE  WALSH   ARITHMETICS.  33 

same  number  of  exercises  being  employed  as  in  multi- 
plication. 

After  nearly  a  half  dozen  pages  of  review  work 
and  "miscellaneous"  problems,  also  some  extension  of  Development 
numeration  and  notation,  pupils  are  led  to  "carrying"   of  Process  by 
in  multiplication  by  being  asked  to  find  products  of     up 
12,  13,  14,  15,  16  and  17  by  2.     Then  come  the  quo- 
tients of  24,  26,  28,  30,  34  and  38  divided  by  2.     It  is 
left  to  the  teacher  to  decide  whether  or  not  she  needs 
to  show  the  "how."     When  the  pupil  has  used  2  as  a 
multiplier  and  a  divisor  in  a  number  of  examples,  he 
is  plunged  into  a  number  of  others  in  which  the  mul-  Commtita- 
tipliers   (or  divisors)    include    numbers    to    9.     The  ^on. 
work,  however,  is  kept  within  his  range  by  using  in 
the  multiplicand  only  (or  quotient)  numbers  composed 
of  O's,  1's  and  2's.     From  this  he  learns  that  he  al- 
ready knows  a  portion  of  the  "3  times"  table,  also  a 
portion  of  each  of  the  others. 

The  child  learns  the  "3  times"  table  by  working 
numerous  examples  in  multiplication  and  division  with 
this  number  as  the  multiplier  or  the  divisor.  He  then, 
as  before,  uses  the  other  numbers  to  9  as  multipliers 
or  divisors,  with  multiplicands  or  quotients  limited  to 
numbers  made  up  of  O's,  1's,  2's  and  3's.  Working 
in  this  way,  he  not  only  learns  each  table  easily  and 
thoroughly,  but  he  begins  to  understand  the  law  of 
commutation,  and  to  realize  that  as  he  goes  towards  9 
times,  he  has  fewer  facts  to  memorize  in  each  table. 

This  lengthy  explanation  of  the  work  of  the  first 
two  chapters  is  intended  to  show  how  much   attention 
has  been  paid  to  making  the  child's  path   in  numbers 
as  smooth  and  as  interesting  as  possible.      The  short-  child?8 
ness  of  the  examples  and  the  care  to  avoid  introducing  Interest. 


34  THE  WALSH  ARITHMETICS. 

difficulties  too  rapidly,  tend  to  give  the  pupil  a  sense 
of  power.  This  obtained,  his  interest  is  secured,  and 
everything  goes  smoothly.  When  a  child  that  has 
used  only  2  as  a  multiplier  is  asked  to  multiply 
121  by  3,  222  by  4,  201  by  5,  121  by  6,  202  by  7, 
112  by  8,  and  212  by  9,  he  is  delighted  to  find  that  the 
new  multipliers  present  no  new  difficulties,  and  he  is 
encouraged  in  his  onward  course. 

In  discussing  the  matter  of  the  second  chapter,  no 
reference  has  been  made  to  the  drills,  oral  and  sight, 
opics.  proDiems  orai  an(j  written;  nor  to  the  new  matters  in- 
troduced— United  States  money,  fractional  parts  of 
numbers,  Roman  notation,  liquid  measure.  Each  turn 
of  the  ''spiral"  brings  in  its  new  matter,  besides  am- 
plifying and  extending  the  old.  It  must  not  be  sup- 
posed, either,  that  abstract  work  constitutes  the  sole 
important  feature  of  the  Walsh  books. 

These  books  lay  as  much  stress  on  the  reasoning 
Accuracy  and  s^e  as  ^°  an^  °ther  good  books;  but  they  also  recog- 
Rapidity.  nize  the  important  fact  that  the  ability  to  reason  cor- 
rectly in  mathematics  is  useless  if  not  accompanied  by 
the  ability  to  compute  accurately.  The  early  school 
years  are  the  ones  to  be  given  to  the  endless  examples 
needed  to  secure  accuracy  and  rapidity  in  performing 
operations,  as  children  at  this  period  are  ready  and 
willing  to  give  themselves  up  to  the  grind  necessary  to 
secure  these  results.  If  they  haven't  mastered  the 
fundamental  processes  before  they  are  11  or  12,  the 
chances  are  that  they  will  always  be  slow  and  inaccur- 
ate. 

The  care  shown  in  the  gradual  development  of  the 

work  contained  in  the  first  and  second  chapters  extends 

Development,  to  all  the  others.     In  long  division,  for  instance,  Chap- 


THE  WALSH   ARITHMETICS.  35 

ter  IV,  none  of  the  first  300  examples  has  a  dividend 
of  over  four  figures,  although  the  multiplication  re- 
sults in  the  same  chapter  generally  contain  five  figures. 
As  long  division  is  rather  difficult,  the  likelihood  of 
the  pupil  becoming  discouraged  is  diminished  by  the 
shortness  of  the  examples,  especially  as  the  earlier 
quotients  consist  of  numbers  containing  small  figures; 
such  as,  13,  21,  22,  23,  12,  211,  123,  222,  11,  12,  etc. 
The  divisors,  too,  are  carefully  chosen;  21,  31,  41,  22, 
32,  42,  etc.,  being  used  before  16.  The  limit  of  four 
figures  in  the  dividends  of  the  early  examples  permits 
of  the  early  introduction  of  large  divisors  without  real- 
ly increasing  the  difficulty  of  the  example,  since  the 
longer  the  divisor  the  fewer  figures  there  will  be  in  the 
quotient;  the  answers  to  8199-i-911  and  9872-^2468 
consisting  of  a  single  figure.  A  word  will  be  said  later 
about  the  long-division  drills. 

A  form  of  fraction  work  is  begun  very  early.  As  Fractions, 
soon  as  children  learn  to  divide  by  2,  they  find  one- 
half  of  a  number.  Later,  they  find  fourths  and  thirds, 
without,  however,  hearing  of  "fraction,"  "numera- 
tor,'* "denominator,"  or  the  like.  The  sum  of  ^  and 
YV,  and  of  1%  and  1^,  etc.,  begins  addition  of  frac- 
tions in  Chapter  III,  although  the  formal  work  is  not 
reached  until  Chapter  VII.  Each'  turn  of  the  *  'spiral' ' 
brings  in  its  appropriate  work  in  Chapters  IV,  V,  and 
VI,  while  each  chapter  after  the  seventh  has  the  need- 
ed reviews. 

Chapter  II  marks  the  commencement  of  work  in  Denominate 
denominate  numbers,  with  problems  involving  pints  Numbers, 
and  quarts.     The  intervening  chapters  to  the  ninth 
extend  the  child's  knowledge  of  this  important  topic, 
while  the  ninth  summarizes  and  completes  the  subject, 


36  THE   WALSH   ARITHMETICS. 

except  for  the  subsequent  inevitable  reviews.  I/ong, 
tedious  examples  are  avoided  by  limiting  the  number 
of  denominate  units  in  any  example  to  two  successive 
units  before  the  ninth  chapter,  and  to  three  successive 

Examples.  units  in  the  ninth  and  remaining  chapters.  This  fea- 
ture of  short  examples  is  a  prominent  one  in  the 
Walsh  books,  and  it  appeals  to  every  one  interested  in 
education.  The  old  time  teacher  that  covered  the 
blackboard  with  a  single  example  in  addition,  for  in- 
stance, did  much  to  kill  the  pupils'  interest  in  mathe- 
matics. A  child  that  is  given  ten  or  a  dozen  short 
examples  during  an  arithmetic  lesson  has  a  chance  of 
getting  the  correct  answer  to  a  large  proportion  of 
them,  while  unable  to  continue  the  strain  needed  to 
work  out  a  single  very  long  one. 

From  the  beginning  of  the  sixth  chapter,  the  point 
at  which  pupils  generally  take  up  a  second  book,  the 
superiority  of  the  (< spiral"  method  becomes  more 
apparent.  In  the  lower  grades,  many  teachers  do 

The  Old-time  good  work  because  they  are  not  hampered  by  *  'logical' ' 
books  in  the  pupils'  hands.  When  the  children  of 
the  5th  year  get  a  book  of  this  kind,  they  are  fettered. 
The  early  pages,  devoted  to  the  fundamental  processes, 
appear  too  elementary,  and  are  not  touched;  while  the 
topical  arrangement  prevents  the  extension,  until  it  is 
regularly  reached,  of  some  work  already  begun.  In 
the  first  chapter  of  the  second  Walsh  book  (Chapter  VI) 
all  the  previous  work  is  continued  and  extended,  while 
new  ground  is  broken  in  decimals  and  mensuration, 
each  treated  in  such  a  way  as  to  be  readily  understood 
by  the  5th  year  scholar.  A  year  later,  Chapter  VIII, 
marks  the  point  at  which  are  introduced  percentage 
and  interest. 


THE  WALSH   ARITHMETICS.  37 

The  employment  of  the  "spiral"  method  does  not 
prevent  the  author  of  the  Walsh  books  from  adopting 
the  good  features  of  other  books.  The  early  intro-  Treatment^ 
duction  of  an  advanced  topic  is  always  accompanied  by  Topic*. 
its  systematic  treatment  in  the  chapter  especially  de- 
voted to  that  topic,  which  chapter  is  reached  in  the 
Walsh  books  at  just  the  same  time  it  is  reached  in  the 
old-line  texts.  Thus,  Chapter  VII  of  Walsh  is  the 
fraction  one;  Chapter  VIII,  the  decimal  one;  Chapter 
IX,  the  denominate  number  one;  Chapter  XI,  the  next 
arithmetical  one,  being  given  to  percentage;  etc.,  etc. 

Besides  being  strong  in  its  general  features,  the 
Walsh  books  are  particularly  useful  for  teaching  pur-  Special 
poses  because  of  their  special  features.  One  marked  I 
characteristic  is  the  space  devoted  to  "drills,"  oral  and 
sight,  each  chapter  containing  its  share.  One  kind  of 
drills  is  intended  to  make  children  masters  of  all  the 
combinations  needed  in  their  work  in  the  fundamental 
processes,  including  two  sets,  never  before  used  in  this 
country,  to  enable  pupils  to  obtain  rapidly  each  figure 
of  the  quotient  in  a  long  division  example.  (See  Art. 
321  and  Arts.  397 — 401.)  These  preliminary  drills  are 
furnished  in  great  variety,  to  prevent  the  weariness  to 
children  that  comes  from  tiresome  repetitions  of  the 
same  exercise. 

As  children  brought  up  on  the  old  books  advanced 
into  more  advanced  topics,  they  seemed  to  lose  their 
earlier  facility  in  computation,  because  of  lack  of  re- 
views, discontinuance  of  drills,  etc.  The  Walsh  books 
aim  not  only  to  keep  up  the  skill  obtained  in  the  lower 
classes  in  adding,  multiplying,  etc. ,  mentally  and  on 
paper,  but  to  increase  it  as  far  as  possible.  A  business 
man  should  not  need  to  hunt  up  a  pencil  every  time  he 


38 


THE   WALSH    ARITHMETICS. 


Special 
Drills. 


wishes  to  make  a  simple  calculation.  f  To  enable  a  boy 
or  a  girl  to  readily  combine  large  numbers,  each  half- 
yearly  chapter  has  a  page  (or  more)  devoted  to  "spec- 
ial drills,"  which  gradually  increase  in  difficulty,  as 
will  be  seen  from  an  examination  of  the  following 
selections: 

FROM  CHAPTER  III. 


50+30 

20+60 

50+40 

40+50 

30+60 

90—50 

50—20 

80—40 

50—30 

90—70 

20X2 

3X30 

20X4 

ysX90 

20X3 

40-5-2 

90-^-30 

y>>  of  60 

80-5-4 

40-5-2 

FROM  CHAPTER  IV. 

13+13  19+30  43+46  51+37  22+23 

25—13  31—20  65—11  87—75  46—26 

13X2  32X3  21X4  23X3  41X2 

88-r-4  39-^13  26-5-2  63-5-21  86-^-2 

FROM  CHAPTER  V. 

56  +  17  13+78  25+16  18+45  34+19 

66—19  56—39  60—12  76—57  43—18 

13X4  5X15  28X3  7X13  47X2 

42-5-3  42-5-14  78n-6x  78-5-13  90-^6 

The  foregoing  types  indicate  the  gradual  develop- 
ment of  the  drills;  the  next  set,  from  Chapter  XIV, 
shows  what  pupils  of  the  eighth  school  year  should 
be  able  to  do  without  using  a  pencil: 


112+91+85 
150—23  +  48 

63X28 
676-T-13 

84Xlf 


43+131+61 

172+   19—66 

54X42 

527^-17 

211^X13 


95  +  144+79 
183— (72— 37) 

26X58 
704-f-22 
36X49| 


THE   WALSH    ARITHMETICS. 

68+56+174 
161  +  79—12 
71X82 
837-^-27 


After  each  set  of  drills,  there  are  given  many  oral 
(or  sight)  problems  involving  similar  combinations,  a 
feature  found  in  no  other  arithmetic,  written  or  mental,  problems 

The  advanced  mental  examples,  even  in  books  devoted  Involving  the 

"SDecieil 
exclusively  to  this   form  of   arithmetic,   use  smaller  Drills." 

numbers  as  the  work  progresses,  the  authors  consid- 
ering, apparently,  that  facility  in  computation  is  of  no 
value  as  compared  with  the  ability  to  crack  mathematic 
chestnuts.  While  the  mental  arithmetic  work  in  the 
Walsh  books  includes  examples  and  problems  such  as 
are  described  above,  at  the  beginning  of  this  paragraph, 
it  also  includes  the  customary  exercises  bearing  upon 
the  topic  under  immediate  treatment. 

Other  sets  of  drills  have  for  their  object  the  devel- 
opment of  skill  in  the  use  of  short  methods  by  pupils. 
They  include  mental  multiplication  and  division  by  commutation 
fractional  parts  of  100;  the  use  of  99,  24,  49,  also  of  Drills. 
99^,  24^,  49^,  etc.,  as  multipliers  in  mental  exam- 
ples, etc.  The  problems  under  this  head,  and,  in  fact, 
problems  all  through  the  books,  show,  where  possible, 
the  law  of  "commutation":  for  instance,  that  25  yards 
at  48c.  per  yard  can  be  solved  mentally  in  the  same 
way  as  48  yards  at  25c.  ;  that  in  subtraction  examples 
the  child  can  take  43  from  50  when  he  can  take  7  from 
50;  that,  in  multiplication,  he  knows  9  fours  when  he 
has  learned  4  nines;  that,  in  division,  if  35  contains  5 
sevens,  it  contains  7  fives. 


40  THE;  WALSH  ARITHMETICS. 

"Approxima-         The  "approximation"  drills  are  new  to  text  books 
tion"  Drills.    jn  arithmetic.     Their  value  is  appreciated  on  sight  by 
all  teachers  anxious  to  prevent  their  pupils  from  offer- 
ing absurd    answers.       The   use   of    the   method  of 
~— approximation  before  attempting  to  solve  a  problem, 
will  frequently  lead  a  pupil  to  discover  the  operations 
necessary  to  its  solution,  whereas,   the  too  frequent 
practice  among  average  children  is  to  begin  to  work 
without  a  full  appreciation  of  the  conditions  involved. 
With  the  purpose  of  making  the  arithmetical  jour- 
ney as  smooth  as  possible  for  young  learners,  the  Walsh 
arithmetics    suggest    some    improvements    upon   the 
method  now  in  vogue.    Teachers  that  believe  children 
should  not  do  anything  without  a  knowledge  of  the 
Method.  underlying  reasons,   have  made  subtraction  unneces- 

sarily difficult  by  requiring  pupils  to  work  examples 
in  the  '  'logical' '  way.  In  finding  the  difference  between 
835  and  398,  the  little  learner  is  supposed  to  make  a 
speech  in  some  such  fashion  as  this:  "Eight  units 
from  5  units  I  cannot  take,  so  I  borrow  1  ten  from  the 
3  tens.  Adding  this  ten,  which  equals  10  units,  to 
the  5  units,  I  have  15  units.  Then  I  take  8  units  from 
15  units  which  leaves  me  7  units,  and  this  I  write  in 
the  units  column.  Since  I  took  1  ten  from  the  3  tens, 
I  have  two  tens  remaining.  As  I  cannot  take  9  tens 
from  2  tens  I  must  borrow  1  hundred  from  the  8  hun- 
dreds. Adding  this  hundred,  which  equals  ten  tens," 
etc.,  etc.;  but  why  continue  this  rigamarole?  And 
how  the  difficulty  is  increased  when  the  minuend 
contains  a  few  ciphers,  say  1000 — 473,  where  the 
1  'next  higher  order"  has  nothing  to  lend.  The  old 
way  is  just  as  "logical":  8  from  15  leaves  7,  10  from 
13  leaves  3,  4  from  8  leaves  4;  but  it  is  more  difficult 


THE  WALSH   ARITHMETICS.  41 

to  "explain".  Is  there,  however,  any  likelihood  that 
infants  understand  the  explanation  they  profess  to  give 
of  the  other  method?  Should  school  children  be  re- 
quifed  to  repeat  an  unmeaning  formula  they  will  never 
use  after  leaving  school?  An  accountant  does  not 
think  of  units,  tens,  or  hundreds  as  he  makes  his  daily 
calculations. 

In  the  Walsh  book,  the  "building  up"  method  is 
advised;  or,  as  it  is  sometimes  called,  the  "computer's  Method  in 
method."  Instead  of  being  told  to  take  8  from  15,  Subtraction, 
the  child  is  asked  "8  and  what  make  15?"  as  expe- 
rience shows  that  the  mind  travels  forward  more  easily 
than  it  goes  backward,  especially  after  giving  all  its 
attention  previously  to  addition.  By  this  method  the 
operation  resembles  addition  so  much  as  to  make  it 
less  difficult  for  beginners.  The  use  of  this  method 
enables  the  pupil  to  shorten  many  other  operations. 
He  can,  for  instance,  ascertain  the  result  of  the  follow- 
ing: 1000— (643+287  +  25)  without  first  finding  the 
sum  of  these  numbers  to  be  added,  (Art.  384);  or 
4832 — (456  X  8) .  Long  division  can  be  performed  with- 
out writing  the  partial  products  (Art.  616);  or  the  mixed 

1 1 223 
number  equivalent  to  can  be  written  at  once. 

In  long  division,  the  children  are  advised  to  write 
each  quotient  figure  above  the  corresponding  figure  of  Quotient, 
the  dividend,  to  prevent  the  omission  of  one  or  more 
ciphers  in  the  quotient,  or  the  introduction  in  the  quo- 
tient of  a  figure  too  many  (Art.  282).  This  plan  is 
similar  to  the  one  used  in  short  division;  and  it  makes 
the  necessary  multiplication  more  easy  to  the  young 
student  by  bringing  the  multiplying  quotient  figure 
nearer  to  the  divisor — an  important  trifle. 


42 


THE   WALSH   ARITHMETICS. 


Division  of 
Decimals. 


Short 
Method. 


Omission  of 
Unnecessary 
Figures. 


The  method  given  for  "pointing-off"  in  division  of 
decimals  (Art.  663)  is  a  mechanical  one,  but  it  pre- 
vents the  pupil  from  getting  the  decimal  point  in  the 
wrong  place.  Another  good  method,  which  helps  by 
mechanical  means  in  getting  the  correct  answer  in 
multiplication,  is  given  in  Art.  344.  The  intelligent 
teacher  will  not  despise  anything  that  will  aid  her 
pupils  to  secure  accurate  results,  even  if  it  is  sneered 
at  as  ''mechanical"  by  the  user  of  "logical"  methods 
whose  pupils  frequently  blunder. 

Besides  giving  much  attention  to  special  short 
methods,  the  Walsh  books  offer  suggestions  in  every 
chapter  as  to  the  disuse  of  unnecessary  figures. 
Children  learn  just  as  readily  to  cipher  without  these 
aids  (?)  as  with  them,  and  their  written  work  becomes 
more  accurate  by  not  being  too  long  drawn  out.  For 
instance,  in  finding  the  least  common  multiple  of  3, 
9,  7,  14,  6,  14,  2,  12,  some  teachers  permit  pupils  to 
retain  all  these  numbers,  instead  of  using  only  the 
necessary  ones — 9,  14,  12  (Art.  595).  In  reducing 
28^  to  an  improper  fraction  or  in  multiplying  16^  by 
8,  scholars  are  not  required,  as  they  should  be,  to  write 
the  answer  directly  (Art.  653). 

To  reduce  15  gal.  3  qt.  to  quarts,  the  average  boy 
or  girl  will  use  several  lines  of  figures,  when  one  is 
sufficient  (Art.  766);  see  also  under  interest  (Art.  936), 
discount  (Art.  937),  commercial  discount  (Art.  944), 
compound  interest  (Art.  983).  To  enumerate  all 
the  places  in  which  suggestions  are  made  as  to  omit- 
ting unneccessary  figures,  would  be  to  make  too  long 
a  list.  The  index  to  the  Grammar  School  Arithmetic 
gives  56  pages  on  which  are  found  "short  methods," 


THE   WALSH    ARITHMETICS.  43 

and  these  do  not  include  suggestions   given  in  the 
Teachers'  Manual. 

The  non-progressive  teacher  hesitates  to  do  things 
in  a  strange  way ;  but  she  will  soon  realize  the  advis- 
ability of  saving  time  as  indicated  in  the  Walsh  books. 
The  methods  given  are  the  straightforward  ones  that 
can  be  understood  by  the  dullest  pupil,  and  which,  by 
being  applicable  in  the  daily  work,  are  readily  appre- 
ciated. They  do  not  include  such  as  9}£  X9}^,  85 X 
85,  64X66,  found  in  some  old  magazine  under  the 
head  of  "Mathematical  Recreations." 

The  algebra  work  is  contained  in  chapters  X  and 
XV;  of  these,  every  pupil  should  study  the  former. 
It  contains  only  1 1  pages,  and  is  readily  understood  Algebra, 
by  a  very  young  pupil.  To  enable  a  teacher  unfami- 
liar with  this  work  to  do  it  successfully,  she  has  only 
to  follow  the  lines  laid  down  in  the  Manual.  While 
chapter  XV  is  intended  more  particularly  for  schools 
that  have  a  nine-years'  course,  its  study  is  advised 
even  if  time  has  to  be  obtained  therefor  by  the  omis- 
sion of  some  of  the  work  in  arithmetic:  bonds  and 
stocks,  for  instance,  compound  interest,  exchange, 
partial  payments,  proportion,  equation  of  payments, 
etc. 

While  the  construction  exercises  and  problems  have 
been  placed  in  chapter  XVI,  they  should  be  commenced 
about  the  time  chapter  XII  is  reached,  and  carried 
along  with  the  arithmetic  work,  even  if  portions  of  the 
latter  receive  less  attention  in  consequence.  The  ex- 
ercises in  calculating  heights  and  distances  are  very  in- 
teresting to  scholars  and  are  very  useful  to  many  of  them 
later  in  life.  They  are  likely  to  be  employed  by  more 
pupils  than  the  problems  in  equation  of  payments. 


44  THE  WAI^SH   ARITHMETICS. 

The  teacher  will  find  in  the  Manual  minute   directions 
as  to  the  best  way  to  conduct  the  geometry  lessons. 

The  Walsh  books  furnish  problems  in  greater  num- 
ber and  variety  than  any  other  series.  Each  problem, 
being  unlike  the  previous  one,  will  require  the  pupil 
to  read  it  carefully;  he  cannot  work  it  by  referring  to 
a  ' 'sample"  one  at  the  head  of  the  page.  The  absurd 
types  are  all  omitted,  such  as  the  far-fetched  ones  in 
some  books  under  the  headings  of  greatest  common 
divisor  and  least  common  multiple. 

These  books  are  offered  to  the  teaching  profession 
in  the  belief  that  they  contain  more  strong  features 
and  are  better  teaching  books  than  any  now  before  the 
public.  It  is  not,  however,  claimed  that  they  are  per- 
fect as  books  of  reference.  Although  an  index  is 
really  unnecessary  in  a  teaching  book,  a  good  one  is 
furnished  for  such  teachers  as  desire  to  use  the  book 
topically. 


The  Walsh  Arithmetics 


C9ntain  abundant,  varied,  and  practical  problems. 
Omit  nothing  essential,  yet  contain  only  the  essentials. 
Are  fresh,  original,  and  well  graded. 
Secure  constant  review  without  actual  repetition. 
Are  arranged  on  the  "  spiral "  plan. 


Three=Book  Series. 

Elementary  Arithmetic. —  For  third  and  fourth  grades. 

Cloth.    2 18  pages.    30  cents 

Intermediate  Arithmetic. —  For  fifth  and  sixth  grades. 

Cloth.    252  pages.    35  cents. 

Higher  Arithmetic. —  For  upper  grades. 

Half  Leather.    387  pages.    65  cents. 


Two-Book  Series. 


Primary  Arithmetic. —  For  third,  fourth  and  fifth  grades. 

doth.     198  pages.    30  cents. 

Grammar  School  Arithmetic. —  For  upper  grades. 

Half  Leather.    433  pages.    65  cents. 


Each  series  is  provided  with  Teacher's  Manuals  in  parts. 
Correspondence  is  cordially  invited. 


D.  C.  HEATH    &    CO.,    Publishers,  Boston,  New  York,  Chicago. 


22  MATHEMATICS. 


Mathematics  for  Common  Schools. 

A  graded  course  in  arithmetic,  with  simple  problems  in  algebra  and  geometry. 
By  JOHN  H.  WALSH,  Associate  Superintendent  of  Public  Instruction,  Brooklyn. 
Two-Book  Series.  —  Primary  Arithmetic.  Cloth,  206  pages.  Introduction 
price,  30  cents.  Grammar  School  Arithmetic.  Half  leather,  458  pages.  Intro- 
duction price,  65  cents. 

Three-Book  Series.—  Elementary  Arithmetic.  Cloth,  220  pages.  Introduc- 
tion price,  30  cents.  Intermediate  Arithmetic.  Cloth,  255  pages.  Introduction 
price,  35  cents.  Higher  Arithmetic.  Half  leather,  403  pages.  Introduction 
price,  65  cents. 

IN  several  important  particulars  the  Walsh  Arithmetics  mark  a  de- 
parture from  the  traditional  method  and  arrangement. 

1.  By  the  "spiral  advancement  plan"  the  elements  of  all  the  im- 
portant topics  are  taken  up  early  in  the  course,  adding  to  the  interest 
and  practical  worth  of  the  study. 

2 .  In  each  case  the  subject  taken  up  is  not  exhausted  at  once,  but 
practice  in  it  is  carried  on  with  problems  of  gradually  increasing  diffi- 
culty throughout  the  course. 

3.  Drills  in  addition,  subtraction,  multiplication  and  division  of  ab- 
stract  numbers   are   given  at  intervals  throughout  the  books  of  the 
series,  thus  insuring  in  pupils  of  the  upper  grades,  accuracy  and  speed 
in  the  fundamental  processes.    This  is  an  important  and  unique  feature. 

4.  The  series  contains  a  larger  number  of  varied  and  practical  con- 
crete problems  than  any  other. 

5.  It  is  the  only  series  containing  drills  in  securing  "approximate 
answers,"  —  work  of  great  advantage  in  calling  the  pupil's  attention 
to  the  condition  of  a  problem,  and  thus  giving  the  power  to  detect  at 
once  the  absurdity  of  any  result  greatly  wide  of  the  mark. 

Such  obvious  merits  of  the  lower  books  as  the  alternation  of  oral, 
sight  and  written  work,  the  early  introduction  of  United  States 
currency  (leading  to  decimals),  the  easy  beginnings  with  fractions  and 
denominate  numbers,  and  the  freshness  and  interest  insured  by  the 
great  variety  of  means  used  to  secure  perfect  mastery  of  simple 
number  combinations,  cannot  be  too  strongly  emphasized. 

In  the  higher  book  are  to  be  noted  the  wide  range  of  subjects  treated 
in  their  simple  elements,  the  great  variety  of  practical  problems,  the 


MA  THEM  A  TICS.  23 


early  introduction  of  percentage  and  simple  interest,  of  bills  and  re- 
ceipts, and  all  the  matters  connected  with  simple  commercial  arithmetic. 

Unique  features  are  :  the  many  short  methods  noted,  the  use  of  ap- 
proximate answers,  the  abundant  drills  in  the  four  fundamental 
processes,  and  the  introduction  of  algebra  in  a  way  so  natural  and  sim- 
ple that  children  of  ten  may  easily  grasp  enough  of  it  to  shorten  many 
of  the  longer  arithmetical  processes. 

The  Walsh  Arithmetics  constitute  a  one-book  series  bound  for  con- 
venience in  two  or  in  three  parts.  The  first  page  of  one  book  follows 
immediately  after  the  last  page  of  the  preceding  one,  without  a  break. 
The  purchaser  of  the  second  book  does  not  buy  a  number  of  useless 
pages,  as  he  must  frequently  do  in  the  case  of  other  series. 

The  Walsh  books  illustrate  most  admirably  what  every  teacher 
knows  so  well,  that  many  things  that  are  complex  in  their  completeness, 
are  in  their  elements  simply  and  easily  comprehended  by  young  chil- 
dren. 

The  series  thoroughly  satisfies  demands  of  modem  pedagogy ;  it  is 
inductive  in  method,  practical  and  varied  in  treatment,  makes  clear 
thought  and  accurate  computation  matters  of  habit,  and  lays  the  foun- 
dation for  the  intelligent  use  of  mathematical  principles. 

The  Walsh  Arithmetics  anticipated  the  recommendations  of  the  Com- 
mittee of  Ten  and  of  the  Committee  of  Fifteen. 

Full  descriptive  circular ',  and  valuable  pamphlets  upon  * '  The  Spiral 
Method"  and  "Suggestions  to  Teachers  and  Courses  of  Study  in 
Arithmetic"  sent  free  on  request. 

Teachers     Manuals   to  Mathematics  for  Common  Scttools. 

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D.C.  HEATH  &  CO.,  Publishers,  Boston, New  York,  Chicago 


THE  WALSH  ARITHMETICS 


SUGGESTIONS  TO  TEACHERS 

AND 

OUTLINES  OF  COURSES  OF  STUDY 


THREE-BOOK   f  E"ementary  Arithmetic  .  .  .  Introduction  price,.  30  cents 
SERIES        l  Interme(^iate  Arithmetic  .  .  **  "     35    " 

I  Higher  Arithmetic «*  "     65     " 

TWO-BOOK     [  Primary  Arithmetic Introduction  price,  30  cents 

SERIES        1  Grammar  School  Arithmetic,  "  »*     65    u 


D.   C.   HEATH   &   CO.,   PUBLISHERS 

BOSTON    NEW  YORK    CHICAGO    LONDON 


CONTENTS.  — PART  I. 


CHAPTER  I. 

PAOB 

ADDITION  AND  SUBTRACTION 1 

NOTATION  AND  NUMERATION  TO  99     ..-.*, 3 

Addition 3 

Subtraction 9 

Addition ...    .  15 

NOTATION  AND  NUMERATION  TO  999 18 

Subtraction 25 

Drills 28 

CHAPTER  II. 

MULTIPLICATION  AND  DIVISION 36 

Multiplication  by  2 36 

Division  by  2 38 

NOTATION  AND  NUMERATION  TO  9,999 42 

Multiplication  by  3 50 

Drills 51 

Division  by  3 ,..-.... 54 

Multiplication  by  4 57 

Division  by  4 59 

Multiplication  by  5 .  62 

Division  by  5 63 

United  States  Money 64 

Addition  and  Subtraction 65 

Fractional  Parts  of  Numbers 67 

Roman  Notation .„ 68 

Liquid  Measure •••*.,».,.  69 


VI  CONTENTS.  —  PART  I. 

CHAPTER   III. 

PAGE 

MULTIPLICATION  AND  DIVISION  —  OUNCE  AND  POUND  —  Two  OPERA- 
TIONS—  HALVES,  THIRDS,   FOURTHS  —  MULTIPLICATION  BY  A 

MIXED  NUMBER 72 

Multiplication  by  6 72 

Division  by  6 75 

Quotients  and  Remainders 80 

Multiplication  by  a  Mixed  Number 81 

NOTATION  AND  NUMERATION  TO  99,999 83 

Multiplication  by  7 85 

Division  by  7 85 

Multiplication  by  8 90 

Division  by  8 90 

Multiplication  by  9 94 

Division  by  9  .     .     .     «, 94 

Multiplication  by  10     ...... 98 

Division  by  10 98 

Special  Drills 99 

Halves  (Addition  and  Subtraction) 105 

Fourths  (Addition  and  Subtraction) .  106 

Thirds  (Addition  and  Subtraction) 107 

CHAPTER   IV. 

MULTIPLIERS  AND  DIVISORS  or  Two  OR  MORE  FIGURES  —  MULTI- 
PLIERS   CONTAINING    FRACTIONS — ADDITION    AND    SUBTRACTION 

OF  EASY  MIXED  NUMBERS  —  INCH,  FOOT.  AND  YARD  .  .  .  110 

Halves  and  Fourths  (Addition  and  Subtraction) 110 

Multiplication  by  11  and  12 113 

Division  by  11  and  12 114 

Multipliers  ending  in  0 118 

Divisors  ending  in  0 118 

Long  Measure 122 

Multipliers  of  Two  Digits , 122 


CONTENTS.  —  PART  I.  vii 

FAGB 

Long  Division 124 

Special  Drills 127 

Halves,  Fourths,  and  Eighths 129 

Multipliers  ending  with  Ciphers 144 

Halves,  Thirds,  and  Sixths 145 

Long  Division  Drills   .    . 148 

Divisors  ending  with  Ciphers 150 

Thirds  and  Ninths    . ,....' 154 

Special  Drills .    .  ".    .    .    .    .    .    .    ...  156 

Multipliers  of  More  than  Two  Figures    ..-..•...•.    .    .  157 

CHAPTER  V. 

MULTIPLIERS  AND  DIVISORS  OP  THREE  OR  MORE  FIGURES — ADDI- 
TION AND  SUBTRACTION  OF  EASY  FRACTIONS — MULTIPLICATION 

BY  A  MIXED  NUMBER — EASY  DENOMINATE  NUMBERS   .    .    .  160 

Multiplication  by  a  Mixed  Number .     .     .  160 

Long  Division 161 

Special  Drills 162 

Mixed  Numbers  (Addition  and  Subtraction)     ........  167 

Dry  Measure 170 

NOTATION  AND  NUMERATION  TO  999,999 172 

More  than  One  Operation 177 

Easy  Fractions  (Addition  and  Subtraction)      .    .    , 179 

Short  Methods ..'.%•    •/•    •     •  181 

Halves  and  Fifths .'  ." 184 

Fourths  and  Fifths 184 

Long  Division,  Drills 186 

Thirds  and  Fourths 192 

Denominate  Numbers 194 

Thirds  and  Fifths     .......    , 201 

Special  DrUls .    .    .    .    v 203 

Roman  Notation          .    . 211 


CONTENTS.  —  PART  II. 


CHAPTER  VI. 

PAGE 

MIXED  NUMBERS  —  FEDERAL  MONEY  —  BILLS  —  DENOMINATE  NUM- 
BERS—  DECIMALS  —  MEASUREMENTS.     .    .    .    .; 213 

MIXED  NUMBERS .  ..    .    .    .... 213 

Addition  of  Mixed  Numbers ,    .  217 

Multiples  and  Factors 218 

Subtraction  of  Mixed  Numbers 220 

NOTATION  AND  NUMERATION 224 

Multiplication  of  Mixed  Numbers 231 

Division  of  Mixed  Numbers  . 234 

FEDERAL  MONEY 236 

Fractional  Parts  of  a  Dollar  .     .     .     .     .     .     .     .     .    .     .    .    .     .  239 

Division  of  Federal  Money 245 

Sight  Approximations i    .    .  247 

DENOMINATE  NUMBERS 250 

Time  Measure ..  . 250 

Dry  Measure .  251 

Avoirdupois  Weight •.,... 251 

Liquid  Measure 251 

Special  Drills  . 254 

BILLS .    .    .    |  .    . 263 

DECIMALS .....*    .    .  264 

Notation  and  Numeration      .     .     .     .    .     .    ....     .     ...     .  265 

Addition  of  Decimals     .    .     .    .....    .    .    .    ...    .     .     .  267 

Subtraction  of  Decimals 268 

Multiplication  of  a  Decimal  by  an  Integer 268 

Division  of  a  Decimal  by  an  Integer 270 

MEASUREMENTS 277 

v 


VI  CONTENTS.  —  PART  II. 

CHAPTER  VII. 

PAGS 

FRACTIONS  —  DECIMALS  —  BILLS  —  DENOMINATE    NUMBERS  —  MEAS- 
UREMENTS   280 

ADDITION  or  FRACTIONS    . * 280 

SUBTRACTION  OF  FRACTIONS 281 

Factors  and  Multiples . 281 

Prime  Numbers 282 

Greatest  Common  Divisor •    • 283 

Lowest  Terms .    .    V •   .    .  283 

Least  Common  Multiple 285 

ADDITION  AND  SUBTRACTION  OF  FRACTIONS 286 

Special  Drills ......    ,    .  291 

Cancellation 296 

MULTIPLICATION  OF  FRACTIONS 298 

DIVISION  OF  FRACTIONS 300 

Fractional  Parts  of  a  Dollar .    .  309 

BILLS 311 

Short  Methods 316 

MULTIPLICATION  OF  DECIMALS 318 

DIVISION  OF  DECIMALS - •.    .    .  319 

Sight  Approximations 322 

DENOMINATE  NUMBERS 327 

Long  Measure 327 

MEASUREMENTS 330 

CHAPTER  VIII. 
DECIMALS  —  BILLS  —  DENOMINATE    NUMBERS  —  MEASUREMENTS  — 

PERCENTAGE  —  INTEREST ...;...  337 

DECIMALS 337 

Reduction '..........  337 

Addition ..."..-..  338 

Subtraction 339 

Multiplication               .    .    .    .    . ;    ;    .  339 

Division  .    , 339 


CONTENTS.  —  PART   II.  Vll 

PAGE 

MEASUREMENTS 345 

Special  Drills 346 

Short  Methods 352 

Approximations .  355 

DENOMINATE  NUMBERS .    .••  .    ;    / 357 

Reduction  Descending 364 

Reduction  Ascending 366 

Addition  and  Subtraction •    •    >    •    :.    •<•"  "."    .     .     .  367 

Multiplication  and  Division   .     .    .    .     ...     .     .     .     .     .     .     .  370 

PERCENTAGE .    .'    ;    .  372 

BILLS >    ''.    .    .    .   \    .    .  376 

INTEREST ;    .    ,    .  377 

AREAS  OF  RIGHT-ANGLED  TRIANGLES -  .    .^  .    .    .  379 

Short  Methods    ,  381 


CHAPTER  IX. 

DENOMINATE  NUMBERS— SURFACES  AND  VOLUMES  —  PERCENTAGE  — 

INTEREST 389 

DENOMINATE  NUMBERS 389 

Reduction  Ascending  and  Descending 389 

Compound  Addition 392 

Compound  Subtraction 394 

Compound  Multiplication 395 

Compound  Division 396 

Special  Drills  .     .    .     ."    ;    =.    ,     .'    .'.    .    ,-.v, ""-.-.    „    .  400 

Short  Methods    ...     .     .'    ,    .     '.    .    .    .    .    .    v    '•: •. . '   .  403 

Avoirdupois  Weight  (Long  Ton) 404 

MEASUREMENTS r  !J  '.    .  408 

Time  between  Dates      .     .    .    . .     .  411 

PERCENTAGE     .    .    .  .."    .    .    .    «    .    . .     .     .  415 

INTEREST. 415 

Approximations 419 


Viii  CONTENTS.  —  PART  II. 

PAGE 

SURFACES 419 

Square  Measure 420 

VOLUMES 424 

Approximations .,...  426 

Cubic  Measure 427 

Troy  Weight 427 

ANGLES,  TRIANGLES,  QUADRILATERALS 443 

Areas 445 

CHAPTER  X. 

ALGEBRAIC  EQUATIONS 447 

ONE  UNKNOWN  QUANTITY 447 

Clearing  of  Fractions 451 

Transposing «455 


CONTENTS.  —  PART  III. 


CHAPTER  XI. 

PAGE 

PERCENTAGE  —  INTEREST  —  DISCOUNT  —  SURFACES  AND  VOLUMES  .     .  459 

PERCENTAGE 459 

To  find  the  Base  or  the  Rate 461 

Profit  and  Loss 464 

MEASUREMENTS 467 

INTEREST 471 

Interest-bearing  Notes  . 472 

Special  Drills 481 

Approximations 483 

Short  Methods .  • 484 

BANK  DISCOUNT 489 

Discount  of  Interest  bearing  Notes 503 

English  Money 504 

COMMERCIAL  DISCOUNT 509 

SURFACES  AND  VOLUMES    .                                                                    .  517 


CHAPTER  XII. 

SIMPLE  AND  COMPOUND  INTEREST  —  DISCOUNT  —  CAUSE  AND  EFFECT  — 
PARTNERSHIP  —  BONDS  AND  STOCKS  —  EXCHANGE  —  LONGITUDE 

AND  TIME  —  SURFACES  AND  VOLUMES 519 

SIMPLE  INTEREST 519 

To  find  Principal,  Rate,  or  Time 519 

Interest  by  Aliquot  Parts 523 

V 


Vi  CONTENTS.  —  PART  III. 

PAGE 

COMMERCIAL  DISCOUNT 529 

BANK  DISCOUNT 534 

To  find  Face  of  Note,  Rate  of  Discount,  or  Time 534 

Special  Drills 537 

Short  Methods 542 

MEASUREMENTS 550 

CAUSE  AND  EFFECT 553 

PARTNERSHIP 558 

Approximations 562 

BONDS  AND  STOCKS 562 

COMPOUND  INTEREST 565 

EXCHANGE 575 

Domestic  Sight  Exchange 576 

Circular  Measure 578 

Time  Drafts 579 

LONGITUDE  AND  TIME 580 

Bills  of  Exchange  (Foreign) 585 


CHAPTER  XIII. 

PARTIAL  PAYMENTS  —  RATIO   AND    PROPORTION  —  SQUARE    ROOT  — 

SURFACES  AND  VOLUMES 588 

PARTIAL  PAYMENTS— U.  S.  RULE 588 

Present  Worth  and  True  Discount 592 

SURFACES  AND  VOLUMES 601 

SQUARE  ROOT 607 

RATIO 610 

Special  Drills 620 

PROPORTION 624 

Applications  of  Square  Root 634 

MEASUREMENTS 643 

Exact  Interest 648 

PARTIAL  PAYMENTS— MERCHANTS'  RULE  .  .  653 


CONTENTS.  —  PART  III.  Vll 
CHAPTER   XIV. 

PAGE 

EQUATION  OF  PAYMENTS — MENSURATION   OF  SURFACES    AND   VOL- 
UMES—  BOARD  MEASURE  —  ANNUAL  INTEREST  —  GOVERNMENT 

LANDS  — METRIC  SYSTEM 657 

EQUATION  OF  PAYMENTS 657 

MENSURATION  OF  PLANE  SURFACES 667 

Special  Drills '.    ....     .    .     .     .  674 

SURFACES  OF  SOLIDS 681 

Prisms  and  Cylinders 681 

Pyramids  and  Cones 682 

VOLUMES .    .    .    .    .    :    ....    .......  688 

Lumber  Measure 692 

Surface  of  Sphere 697 

CUBE  ROOT '.    .    .    .    .  699 

Volume  of  Sphere 701 

ANNUAL  INTEREST     . ,    .  719 

Government  Lands .  720 

METRIC  SYSTEM    .  721 


CHAPTER  XV. 

ALGEBRAIC  EQUATIONS  —  Two  UNKNOWN  QUANTITIES  —  THREE  UN- 
KNOWN QUANTITIES  —  PURE  QUADRATICS  —  AFFECTED  QUAD- 
RATICS .  ....  .  ......  .  .  ...  .  v.  .  .  728 

ADDITION  OF  ALGEBRAIC  QUANTITIES ,    .  -  .  728 

SUBTRACTION  OF  ALGEBRAIC  QUANTITIES .  730 

Removing  Parentheses ' -   .     .     .  732 

Two  UNKNOWN  QUANTITIES .     .  736 

• 

THREE  UNKNOWN  QUANTITIES .-  .,  .    .    .    •<    .    .    .  742 

MULTIPLICATION  OF  ALGEBRAIC  QUANTITIES 745 

PURE  QUADRATICS 747 

AFFECTED  QUADRATICS 749 


Vlll  CONTENTS.  —  PART  III. 

CHAPTER  XVI. 

PAQI 

ELEMENTARY  GEOMETRT — PROBLEMS  IN  CONSTRUCTION  —  PRACTICAL 
APPLICATIONS — CALCULATION  OF  HEIGHTS  AND  DISTANCES — 

MENSURATION 755 

ELEMENTARY  GEOMETRY 755 

Exercises  in  Construction 757 

Problems  in  Construction 773 

Equal  Triangles  — Equivalent  Triangles 782 

Similar  Triangles - 784 

CALCULATION  OF  HEIGHTS  AND  DISTANCES.    .........  785 

MENSURATION  OF  SURFACES 790 

Prisms,  Cylinders,  Pyramids,  Cones 792 

Frustum  of  Pyramid  or  Cone      ..............  794 

Sphere .    .  796 

VOLUMES 797 

Prisms  and  Cylinders 797 

Pyramids  and  Cones ...........  798 

Frustums  of  Pyramids  and  Cones 799 

Oblique  Prisms 802 

Sphere 802 


SUGGESTIONS  TO   TEACHERS 


INTRODUCTORY 

Plan  and  Scope  of  the  Work,  —  In  addition  to  the  subjects 
generally  included  in  text-books  in  arithmetic,  The  Walsh  Arith- 
metics contain  such  simple  work  in  algebraic  equations  and  con- 
structive geometry  as  can  be  studied  to  advantage  by  pupils  of 
the  elementary  schools.  The  arithmetical  portion  is  divided 
into  thirteen  chapters,  each  of  which,  except  the  first,  contains 
a  half-year's  work.  The  following  nine-year  and  eight-year 
courses  will  show  the  arrangement  of  topics  : 

NINE-YEAR  COURSE 
FIRST  AND  SECOND  YEARS 
Chapter  I,  —  Numbers  of  Three  Figures.    Simple  Processes. 

THIRD  YEAR 

Chapters  II,  and  IIL — Numbers  of  Five  Figures.  Multipliers 
and  Divisors  of  One  Figure.  Addition  and  Subtraction  of  Halves,  of 
Fourths,  of  Thirds.  Multiplication  by  Mixed  Numbers.  Pint,  Quart, 
and  Gallon;  Ounce  and  Pound.  Roman  Notation. 

FOURTH  YEAR 

Chapters  IV,  and  V,  —  Numbers  of  Six  Figures.  Multipliers  and 
Divisors  of  Two  or  More  Figures.  Addition  and  Subtraction  of  Easy 
Fractions.  Multiplication  by  Mixed  Numbers.  Simple  Denominate 
Numbers.  Roman  Notation. 

FIFTH  YKAK 

Chapters  VI,  and  VII,  —  Fractions.  Decimals  of  Three  Places. 
Bills.  Denominate  Numbers.  Simple  Measurements. 

SIXTH  YKAU 

Chapters  VIII.  and  IX,  —  Decimals.  Bills.  Denominate  Num- 
bers. Surfaces  and  Volumes.  Percentage  and  Interest. 

1 


2  SUGGESTIONS   TO   TEACHERS 

SEVENTH  YEAR 

Chapters  X,  and  XI,  and  Articles  931  to  963  in  Chapter  XII,, 
and  Articles  1251  to  1269  in  Chapter  XVI,  —  Percentage.  Meas- 
urements. Interest.  Discount.  Surfaces  and  Volumes.  Elementary 
Algebra  and  Geometry.  Exercises  and  Problems. 

EIGHTH  YEAR 

Chapter  XII,,  Articles  964  to  1007,  Chapter  XIII,,  and  Article 
1270  of  Chapter  XVI,  —  Partnership.  Bonds  and  Stocks.  Com- 
pound Interest.  Exchange.  Longitude  and  Time.  Partial  Pay- 
ments. Surfaces  and  Volumes.  Square  Root.  Ratio.  Proportion. 
Measurements.  Elementary  Geometry.  Problems  in  Construction. 

NINTH  YEAR 

Chapters  XIV,  and  XV,,  and  Chapter  XVI,  completed,  —  Equa- 
tion of  Payments.  Mensuration  of  Plane  Surfaces  and  Volumes. 
Cube  Root.  Annual  Interest.  Metric  System.  Elementary  Algebra. 
Elementary  Geometry.  Calculation  of  Heights  and  Distances. 

EIGHT- YEAR  COURSE 
FIRST,  SECOND,  THIRD,  AND  FOURTH  YEARS 

As  in  nine-year  course. 

FIFTH  YEAR 

Chapters  VI,  and  VII,  —  Fractions.  Decimals  of  Three  Places. 
Bills.  Denominate  Numbers.  Simple  Measurements. 

SIXTH  YEAR 

Chapters  VIII,  and  IX,  —  Decimals.  Bills.  Denominate  Num- 
bers. Surfaces  and  Volumes.  Percentage  and  Interest. 

SEVENTH  YEAR 

Chapters  XI,  and  XII,  —  Percentage  and  Interest.  Commercial 
and  Bank  Discount.  Cause  and  Effect.  Partnership.  Bonds  and 
Stocks.  Exchange.  Longitude  and  Time.  Surfaces  and  Volumes. 

EIGHTH  YEAR 

Chapters  XIII,  and  XIV,  —  Partial  Payments.  Equation  of  Pay- 
ments. Annual  Interest.  Metric  System.  Evolution  and  Involution. 
Surfaces  and  Volumes. 


INTRODUCTORY  3 

While  all  of  the  above  topics  are  generally  included  in  an 
eight  years'  course,  it  may  be  considered  advisable  to  omit  some 
of  them,  and  to  take  up,  instead,  during  the  seventh  and  eighth 
years,  the  constructive  geometry  work  of  Chapter  XVI.  Among 
the  topics  that  may  be  dropped  without  injury  to  the  pupil  are 
Bonds  and  Stocks,  Exchange,  Partial  Payments,  and  Equation 
of  Payments. 

Grammar  School  Algebra.  —  Chapter  X.,  consisting  of  a  dozen 
pages,  is  devoted  to  the  subject  of  easy  equations  of  one  unknown 
quantity,  as  a  preliminary  to  the  employment  of  the  equation  in 
so  much  of  the  subsequent  work  in  arithmetic  as  is  rendered 
more  simple  by  this  mode  of  treatment.  To  teachers  desirous 
of  dispensing  with  rules,  sample  solutions  of  type  examples,  etc., 
the  algebraic  method  of  solving  the  so-called  "  problems  "  in  per- 
centage, interest,  discount,  etc.,  is  strongly  recommended. 

In  Chapter  XV.,  intended  chiefly  for  schools  having  a  nine 
years'  course,  the  algebraic  work  is  extended  to  cover  simple 
equations  containing  two  or  more  unknown  quantities,  and  pure 
and  affected  quadratic  equations  of  one  unknown  quantity. 

No  attempt  has  been  made  in  these  two  chapters  to  treat 
algebra  as  a  science ;  the  aim  has  been  to  make  grammar-school 
pupils  acquainted,  to  some  slight  extent,  with  the  great  instru- 
ment of  mathematical  investigation,  —  the  equation. 

Constructive  Geometry,  —  Progressive  teachers  will  appreciate  the 
importance  of  supplementing  the  concrete  geometrical  instruction 
now  given  in  the  drawing  and  mensuration  work.  Chapter  XVI. 
contains  a  series  of  problems  in  construction  so  arranged  as  to 
enable  pupils  to  obtain  for  themselves  a  working  knowledge  of 
all  the  most  important  facts  of  geometry.  Applications  of  the 
facts  thus  ascertained,  are  made  to  the  mensuration  of  surfaces 
and  volumes,  the  calculation  of  heights  and  distances,  etc.  No 
attempt  is  made  to  anticipate  the  work  of  the  high-school  by 
teaching  geometry  as  a  science. 


4  MANUAL   FOR  TEACHERS 

While  the  construction  problems  are  brought  together  into  a 
single  chapter  at  the  end  of  the  book,  it  is  not  intended  that 
instruction  in  geometry  should  be  delayed  until  the  preceding 
work  is  completed.  Chapter  XVI.  should  be  commenced  not  later 
than  the  seventh  year,  and  should  be  continued  throughout  the 
remainder  of  the  grammar-school  course.  For  the  earlier  years, 
suitable  exercises  in  the  mensuration  of  the  surfaces  of  triangles 
and  quadrilaterals,  and  of  the  volumes  of  right  parallelopipedons 
have  been  incorporated  with  the  arithmetic  work. 


II 

GENERAL  HINTS 

Division  of  the  Work,  —  The  five  chapters  constituting  Part  I. 
of  Mathematics  for  Common  Schools  should  be  completed  by  the 
end  of  the  fourth  school  year.  The  remaining  eight  arithmetic 
chapters  constitute  half-yearly  divisions  for  the  second  four  years 
of  school.  Chapter  L,  with  the  additional  oral  work  needed  in 
the  case  of  young  pupils,  will  occupy  about  two  years ;  the  re- 
maining four  chapters  should  not  take  more  than  half  a  year  each. 
When  the  Grube  system  is  used,  and  the  work  of  the  first  two 
years  is  exclusively  oral,  it  will  be  possible,  by  omitting  much  of 
the  easier  portions  of  the  first  two  chapters,  to  cover,  during  the 
third  year,  the  ground  contained  in  Chapters  I.,  II.,  and  III. 

Additions  and  Omissions.  —  The  teacher  should  freely  supple- 
ment the  work  of  the  text-book  when  she  finds  it  necessary  to  do 
so ;  and  she  should  not  hesitate  to  leave  a  topic  that  her  pupils 
fully  understand,  even  though  they  may  not  have  worked  all  the 
examples  given  in  connection  therewith.  A  very  large  number 
of  exercises  is  necessary  for  such  pupils  as  can  devote  a  half-year 
to  the  study  of  the  matter  furnished  in  each  chapter.  In  the 
case  of  pupils  of  greater  maturity,  it  will  be  possible  to  make 
more  rapid  progress  by  passing  to  the  next  topic  as  soon  as  the 
previous  work  is  fairly  well  understood. 

Oral  and  Written  Work.  — The  heading  "Slate  Problems"  is 
merely  a  general  direction,  and  it  should  be  disregarded  by  the 
teacher  when  the  pupils  are  able  to  do  the  work  "  mentally." 
The  use  of  the  pencil  should  be  demanded  only  so  far  as  it  mny 

5 


6  MANUAL   FOR   TEACHERS 

be  required.  It  is  a  pedagogical  mistake  to  insist  that  all  of  tlie 
pupils  of  a  class  should  set  down  a  number  of  figures  that  are 
not  needed  by  the  brighter  ones.  As  an  occasional  exercise,  it 
may  be  advisable  to  have  scholars  give  all  the  work  required  to 
solve  a  problem,  and  to  make  a  written  explanation  of  each  step 
in  the  solution ;  but  it  should  be  the  teacher's  aim  to  have  the 
majority  of  the  examples  done  with  as  great  rapidity  as  is  con- 
sistent with  absolute  correctness.  It  will  be  found  that,  as  a 
rule,  the  quickest  workers  are  the  most  accurate. 

Many  of  the  slate  problems  can  be  treated  by  some  classes  as 
"  sight "  examples,  each  pupil  reading  the  question  for  himself 
from  the  book,  and  writing  the  answer  at  a  given  signal  without 
putting  down  any  of  the  work. 

Use  of  Books,  —  It  is  generally  recommended  that  books  be 
placed  in  pupils'  hands  as  early  as  the  third  school  year.  Since 
many  children  are  unable  at  this  stage  to  read  with  sufficient 
intelligence  to  understand  the  terms  of  a  problem,  this  work 
should  be  done  under  the  teacher's  direction,  the  latter  reading 
the  questions  while  the  pupils  follow  from  their  books.  In  later 
years,  the  problems  should  be  solved  by  the  pupils  from  the 
books  with  practically  no  assistance  whatever  from  the  teacher. 

Conduct  of  the  Eecitation,  —  Many  thoughtful  educators  consider 
it  advisable  to  divide  an  arithmetic  class  into  two  sections,  for 
some  purposes,  even  where  its  members  are  nearly  equal  in 
attainments.  The  members  of  one  division  of  such  a  class  may 
work  examples  from  their  books  while  the  others  write  the 
answers  to  oral  problems  given  by  the  teacher,  etc. 

Where  a  class  is  thus  taught  in  two  divisions,  the  members  of 
each  should  sit  in  alternate  rows,  extending  from  the  front 
of  the  room  to  the  rear.  Seated  in  this  way,  a  pupil  is  doing  a 
different  kind  of  work  from  those  on  the  right  and  the  left,  and 
he  would  not  have  the  temptation  of  a  neighbor's  slate  to  lead 
him  to  compare  answers. 


GENERAL   HINTS  7 

As  an  economy  of  time,  explanations  of  new  subjects  might  be 
given  to  the  whole  class;  but  much  of  the  arithmetic  work 
should  be  done  in  "sections,"  one  of  which  is  under  the  im- 
mediate direction  of  the  teacher,  the  other  being  employed 
in  "seat"  work.  In  the  case  of  pupils  of  the  more  advanced 
classes,  "seat"  work  should  consist  largely  of  "problems"  solved 
without  assistance.  Especial  pains  have  been  taken  to  so  grade 
the  problems  as  to  have  none  beyond  the  capacity  of  the  average 
pupil  that  is  willing  to  try  to  understand  its  terms.  It  is  not 
necessary  that  all  the  members  of  a  division  should  work  the 
same  problems  at  a  given  time,  nor  the  same  number  of  prob- 
lems, nor  that  a  new  topic  should  be  postponed  until  all  of  the 
previous  problems  have  been  solved. 

Whenever  it  is  possible,  all  of  the  members  of  the  division 
working  under  the  teacher's  immediate  direction  should  take 
part  in  all  the  work  done.  In  mental  arithmetic,  for  instance, 
while  only  a  few  may  be  called  upon  for  explanations,  all  of  the 
pupils  should  write  the  answers  to  each  question.  The  same  is 
true  of  much  of  the  sight  work,  the  approximations,  some  of  the 
special  drills,  etc. 

Drills  and  Sight  Work.  —  To  secure  reasonable  rapidity,  it  is 
necessary  to  have  regular  systematic  drills.  They  should  be 
employed  daily,  if  possible,  in  the  earlier  years,  but  should  never 
last  longer  than  five  or  ten  minutes.  Various  kinds  are  sug- 
gested, such  as  sight  addition  drills,  in  Arts.  3,  11,  24,  26,  etc. ; 
subtraction,  in  Arts.  19,  50,  53,  etc. ;  multiplication,  in  Arts.  71, 
109,  etc. ;  division,  in  Arts.  199,  202,  etc. ;  counting  by  2's,  3's, 
etc.,  in  Art.  61 ;  carrying,  in  Art.  53,  etc.  For  the  young  pupil, 
those  are  the  most  valuable  in  which  the  figures  are  in  his  sight, 
and  in  the  position  they  occupy  in  an  example ;  see  Arts.  3,  34, 
164,  etc. 

Many  teachers  prepare  cards,  each  of  which  contains  one  of 
the  combinations  taught  in  their  respective  grades.  Showing 
one  of  these  cards,  the  teacher  requires  an  immediate  answer 


8  MANUAL   FOR  TEACHERS 

from  a  pupil.  If  his  reply  is  correct,  a  new  card  is  shown  to 
the  next  pupil,  and  so  on.  Other  teachers  write  a  number  of 
combinations  on  the  blackboard,  and  point  to  them  at  random, 
requiring  prompt  answers.  When  drills  remain  on  the  board 
for  any  considerable  time,  some  children  learn  to  know  the 
results  of  a  combination  by  its  location  on  the  board,  so  that 
frequent  changes  in  the  arrangement  of  the  drills  are,  therefore, 
advisable.  The  drills  in  Arts.  Ill,  112,  and  115  furnish  a  great 
deal  of  work  with  the  occasional  change  of  a  single  figure. 

For  the  higher  classes,  each  chapter  contains  appropriate 
drills,  which  are  subsequently  used  in  oral  problems.  It  happens 
only  too  frequently  that  as  children  go  forward  in  school  they 
lose  much  of  the  readiness  in  oral  and  written  work  they 
possessed  in  the  lower  grades,  owing  to  the  neglect  of  their 
teachers  to  continue  to  require  quick,  accurate  review  work  in 
the  operations  previously  taught.  These  special  drills  follow 
the  plan  of  the  combinations  of  the  earlier  chapters,  but  gradu- 
ally grow  more  difficult.  They  should  first  be  used  as  sight 
exercises,  either  from  the  books  or  from  the  blackboard. 

To  secure  valuable  results  from  drill  exercises,  the  utmost 
possible  promptness  in  answers  should  be  insisted  upon. 

Definitions,  Principles,  and  Eules,  —Young  children  should  not 
memorize  rules  or  definitions.  They  should  learn  to  add  by 
adding,  after  being  first  shown  by  the  teacher  how  to  perform 
the  operation.  Those  not  previously  taught  by  the  Grube 
method  should  be  given  no  reason  for  "  carrying."  In  teaching 
such  children  to  write  numbers  of  two  or  three  figures,  there  is 
nothing  gained  by  discussing  the  local  value  of  the  digits.  Dur- 
ing the  earlier  years,  instruction  in  the  art  of  arithmetic  should 
be  given  with  the  least  possible  amount  of  science.  While  prin- 
ciples may  be  incidentally  brought  to  the  view  of  the  children 
at  times,  there  should  be  no  cross-examination  thereon.  It  may 
be  shown,  for  instance,  that  subtraction  is  the  reverse  of  addition, 
and  that  multiplication  is  a  short  method  of  combining  equal 


GENERAL   HINTS  9 

numbers,  etc. ;  but  care  should  be  taken  in  the  case  of  pupils 
below  about  the  fifth  school  year  not  to  dwell  long  on  this  side 
of  the  instruction.  By  that  time,  pupils  should  be  able  to  add, 
subtract,  multiply,  and  divide  whole  numbers ;  to  add  and  sub- 
tract simple  mixed  numbers,  and  to  use  a  mixed  number  as  a 
multiplier  or  a  multiplicand  ;  to  solve  easy  problems,  with  small 
numbers,  involving  the  foregoing  operations  and  others  contain- 
ing the  more  commonly  used  denominate  units.  Whether  or  not 
they  can  explain  the  principles  underlying  the  operations  is  of 
next  to  no  importance,  if  they  can  do  the  work  with  reasonable 
accuracy  and  rapidity. 

When  decimal  fractions  are  taken  up,  the  principles  of  Arabic 
notation  should  be  developed  ;  and  about  the  same  time,  or  some- 
what later,  the  principles  upon  which  are  founded  the  operations 
in  the  fundamental  processes,  can  be  briefly  discussed. 

Definitions  should  in  all  cases  be  made  by  the  pupils,  their 
mistakes  being  brought  out  by  the  teacher  through  appropriate 
questions,  criticisms,  etc.  Systematic  work  under  this  head 
should  be  deferred  until  at  least  the  seventh  year. 

The  use  of  unnecessary  rules  in  the  higher  grades  is  to  be 
deprecated.  When,  for  instance,  a  pupil  understands  that  per 
cent  means  hundredths,  that  seven  per  cent  means  seven  hun- 
dredths,  it  should  «ot  be  necessary  to  tell  him  that  7  per  cent  of 
143  is  obtained  by  multiplying  143  by  .07.  It  should  be  a  fair 
assumption  that  his  previous  work  in  the  multiplication  of 
common  and  of  decimal  fractions  has  enabled  him  to  see  that 
7  per  cent  of  143  is  -j-J^  of  143  or  143  X  .07,  without  information 
other  than  the  meaning  of  the  term  "  per  cent." 

When  a  pupil  is  able  to  calculate  that  15%  of  120  is  18,  he 
should  be  allowed  to  try  to  work  out  for  himself,  without  a  rule, 
the  solution  of  this  problem :  18  is  what  per  cent  of  120  ?  or  of 
this:  18  is  15%  of  what  number?  These  questions  should 
present  no  more  difficulty  in  the  seventh  year  than  the  following 
examples  in  the  fifth  :  (a)  Find  the  cost  of  -fo  ton  of  hay  at  $12 
per  ton.  (b)  When  hay  is  worth  $12  per  ton,  what  part  of  a 


10  MANUAL   FOR   TEACHERS 

ton  can  be  bought  for  $  1.80  ?  (c)  If  ^  ton  of  hay  costs  $1.80, 
what  is  the  value  of  a  ton  ? 

When,  however,  it  becomes  necessary  to  assist  pupils  in  the 
solution  of  problems  of  this  class,  it  is  more  profitable  to  furnish 
them  with  a  general  method  by  the  use  of  the  equation,  than 
with  any  special  plan  suited  only  to  the  type  under  immediate 
discussion. 

In  the  supplement  to  the  Manual  will  be  found  the  usual  defini- 
tions, principles,  and  rules,  for  the  teacher  to  use  in  such  a  way 
as  her  experience  shows  to  be  best  for  her  pupils.  The  rules 
given  are  based  somewhat  on  the  older  methods,  rather  than  on 
those  recommended  by  the  author.  He  would  prefer  to  omit 
entirely  those  relating  to  percentage,  interest,  and  the  like  as 
being  unnecessary,  but  that  they  are  called  for  by  many  success- 
ful teachers,  who  prefer  to  continue  the  use  of  methods  which 
they  have  found  to  produce  satisfactory  results. 

Language,  —  While  the  use  of  correct  language  should  be 
insisted  upon  in  all  lessons,  children  should  not  be  required  in 
arithmetic  to  give  all  answers  in  "  complete  sentences."  Espe- 
cially in  the  drills,  it  is  important  that  the  results  be  expressed 
in  the  fewest  possible  words. 

Analyses,  —  Sparing  use  of  analyses  is  recommended  for  begin- 
ners. If  a  pupil  solves  a  problem  correctly,  the  natural  inference 
should  be  that  his  method  is  correct,  even  if  he  be  unable  to  state 
it  in  words.  When  a  pupil  gives  the  analysis  of  a  problem,  he 
should  be  permitted  to  express  himself  in  his  own  way.  Set 
forms  should  not  be  used  under  any  circumstances. 

Objective  Illustrations,  —  The  chief  reason  for  the  use  of  objects 
in  the  study  of  arithmetic  is  to  enable  pupils  to  work  without 
them.  While  counters,  weights  and  measures,  diagrams,  or  the 
like  are  necessary  at  the  beginning  of  some  topics,  it  is  important 
to  discontinue  their  use  as  soon  as  the  scholar  is  able  to  proceed 
without  their  aid. 


GENERAL   HINTS  11 

Approximate  Answers.  —  An  important  drill  is  furnished  in 
the  "approximations."  (See  Arts.  521,  669,  719,  etc.)  Pupils 
should  be  required  in  much  of  their  written  work  to  estimate 
the  result  before  beginning  to  solve  a  problem  with  the  pencil. 
Besides  preventing  an  absurd  answer,  this  practice  will  also  have 
the  effect  of  causing  a  pupil  to  see  what  processes  are  necessary. 
In  too  many  instances,  work  is  commenced  upon  a  problem  before 
the  conditions  are  grasped  by  the  youthful  scholar ;  which  will 
be  less  likely  to  occur  in  the  case  of  one  who  has  carefully 
"estimated"  the  answer.  The  pupil  will  frequently  find,  also, 
that  he  can  obtain  the  correct  result  without  using  his  pencil 
at  all. 

Indicating  Operations,  —  It  is  a  good  practice  to  require  pupils 
to  indicate  by  signs  all  of  the  processes  necessary  to  the  solution 
of  a  problem,  before  performing  any  of  the  operations.  This  fre- 
quently enables  a  scholar  to  shorten  his  work  by  cancellation,  etc. 
In  the  case  of  problems  whose  solution  requires  tedious  processes, 
some  teachers  do  not  require  their  pupils  to  do  more  than  to 
indicate  the  operations.  It  is  to  be  feared  that  much  of  the  lack 
of  facility  in  adding,  multiplying,  etc.,  found  in  the  pupils  of 
the  higher  classes  is  due  to  this  desire  to  make  work  pleasant. 
Instead  of  becoming  more  expert  in  the  fundamental  operations, 
scholars  in  their  eighth  year  frequently  add,  subtract,  multiply, 
and  divide  more  slowly  and  less  accurately  than  in  their  fourth 
year  of  school. 

Paper  vs.  Slates,  —  To  the  use  of  slates  may  be  traced  very  much 
of  the  poor  work  now  done  in  arithmetic.  A  child  that  finds  the 
sum  of  two  or  more  numbers  by  drawing  on  his  slate  the  number 
of  strokes  represented  by  each,  and  then  counting  the  total,  will 
have  to  adopt  some  other  method  if  his  work  is  done  on  material 
that  does  not  permit  the  easy  obliteration  of  the  tell-tale  marks. 
When  the  teacher  has  an  opportunity  to  see  the  number  of 
attempts  made  by  some  of  her  pupils  to  obtain  the  correct  quo- 


12  MANUAL   FOR  TEACHERS 

tient  figures  in  a  long  division  example,  she  may  realize  the 
importance  of  such  drills  as  will  enable  them  to  arrive  more 
readily  at  the  correct  result. 

The  unnecessary  work  now  done  by  many  pupils  will  be  very 
much  lessened  if  they  find  themselves  compelled  to  dispense  with 
the  "  rubbing  out "  they  have  an  opportunity  to  indulge  in  when 
slates  are  employed.  The  additional  expense  caused  by  the 
introduction  of  paper  will  almost  inevitably  lead  to  better  results 
in  arithmetic.  The  arrangement  of  the  work  will  be  looked 
after ;  pupils  will  not  be  required,  nor  will  they  be  permitted,  to 
waste  material  in  writing  out  the  operations  that  can  be  per- 
formed mentally ;  the  least  common  denominator  will  be  deter- 
mined by  inspection  ;  problems  will  be  shortened  by  the  greater 
use  of  cancellation,  etc.,  etc.  Better  writing  of  figures  and  neater 
arrangement  of  problems  will  be  likely  to  accompany  the  use  of 
material  that  will  be  kept  by  the  teacher  for  the  inspection  of 
the  school  authorities.  The  endless  writing  of  tables  and  the 
long,  tedious  examples  now  given  to  keep  troublesome  pupils 
from  bothering  a  teacher  that  wishes  to  write  up  her  records, 
will,  to  some  extent,  be  discontinued  when  slates  are  no  longer 
used. 


The   Walsh   Arithmetics. 


IN  several  important  particulars  the  Walsh  Arithmetics  mark  a 
departure  from  the  traditional  method  and  arrangement. 

1.  By  the  "spiral  plan"  the  elements  of  all  the  important 
topics  are  taken  up  early  in  the  course,  adding  to  the  interest 
and  practical  worth  of  the  study. 

2.  In  each  case  the  subject  taken  up  is  not  exhausted  at 
once,  but  practice  in  it  is  carried  on  with  problems  of  gradually 
increasing  difficulty  throughout  the  course. 

3.  Drills  in  addition,  subtraction,  multiplication  and  division 
of  abstract  numbers  are  given  at  intervals  throughout  the  books 
of  the  series,  thus  insuring  in  pupils  of  the  upper  grades,  ac- 
curacy and  speed  in  the  fundamental  processes.     This  is  an  im- 
portant and  unique  feature. 

4.  The  series  contains  a  larger  number  of  varied  arid  practi- 
cal concrete  problems  than  any  other. 

5.  It  is  the  only  series  containing  drills  in  securing  "  approxi- 
mate answers,"  —  work  of  great  advantage  in  calling  the  pupil's 
attention  to  the  conditions  of  a  problem,  and  thus  giving  the 
power  to  detect  at  once  the  absurdity  of  any  result  greatly  wide 
of  the  mark. 

Such  obvious  merits  of  the  lower  book  as  the  alternation  of 
oral,  sight  and  slate  work,  the  early  introduction  of  United  States 
currency  (leading  to  decimals),  the  easy  beginnings  with  frac- 
tions and  denominate  numbers,  and  the  freshness  and  interest 


insured  by  the  great  variety  of  means  used  to  secure  perfect 
mastery  of  simple  number  combinations,  cannot  be  too  strongly 
emphasized. 

In  the  higher  book  we  note  the  wide  range  of  subjects  treated 
in  their  simple  elements,  the  great  variety  of  practical  problems, 
the  early  introduction  of  percentage  and  simple  interest,  of  bills 
and  receipts,  and  all  the  matters  connected  with  simple  commer- 
cial arithmetic. 

Unique  features  are  :  the  many  short  methods  noted,  the  use  of 
approximate  answers,  the  abundant  drills  in  the  four  fundamental 
processes,  and  the  introduction  of  algebra  in  a  way  so  natural 
and  simple  that  children  of  ten  may  easily  grasp  enough  of  it  to 
shorten  many  of  the  longer  arithmetical  processes. 

The  Walsh  books  illustrate  most  admirably  what  every  teacher 
knows  so  well,  that  many  things  that  are  complex  in  their  com- 
pleteness, are  in  their  elements  simple  and  easily  comprehended 
by  young  children. 

The  series  is  thoroughly  up  to  the  demands  of  modern  peda- 
gogy; it  is  inductive  in  method,  practical  and  varied  in  .treat- 
ment, and  pursues  one  object  from  start  to  finish,  i.  e.,  to  make 
clear  thought  and  accurate  computation  matters  of  habit,  and  to 
lay  the  foundation  for  the  intelligent  use  of  mathematical  princi- 
ples. 

A  comparison  of  the  number  of  subjects  treated  in  any  given 
one  hundred  pages  of  Walsh  with  a  corresponding  one  hundred 
pages  in  any  other  series  makes  evident  Walsh's  superiority  both 
in  variety  and  freshness,  and  in  drill  and  review  upon  essentials. 


The  Heart  of  Oak  Books. 


A  collection  of  traditional  rhymes  and  stories  for  children,  and  of  mas- 
terpieces of  poetry  and  prose,  for  use  at  school  and  at  home,  chosen  with 
special  reference  to  the  cultivation  of  the  imagination  and  a  taste  for  good 
reading.  By 

CHARLES  ELIOT  NORTON. 

These  six  volumes  provide  an  unrivaled  means  of  making  good  reading 
more  attractive  than  bad,  and  of  giving  right  direction  to  uncritical  choice, 
by  offering  to  the  young,  without  comment  or  lesson-book  apparatus, 

SELECTED  PORTIONS  OF  THE  BEST  LITERATURE,  THE  VIRTUE  OK 
WHICH  HAS  BEEN  APPROVED  BY  IX>NG  CONSENT. 

The  selections  are  of  unusual  length,  completeness  and  variety,  compris- 
ing a  very  large  proportion  of  poetry,  and  are  adapted  to  the  progressive 
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THE    WALSH 
ARITHMETICS 


WHAT  THEY  ARE  AND 
WHAT  THEY  WILL  DO 

ALSO 

WHAT  THEY  HAVE  DONE 
AND  ARE  DOING,  TOLD  BY 
THOSE  WHO  USE  THEM 


BOSTON 
ATLANTA 


PUBLISHED  BY 
D.  C.  HEATH  &  COMPANY 

NEW  YORK 
SAN  FRANCISCO 


CHICAGO 
LONDON 


MATHEMATICS 
FOR  COMMON    SCHOOLS 

A  Graded  Course  in  Arithmetic  with 

Simple  Problems  in  Algebra 

and  Geometry 

By  JOHN  H.  WALSH 

Associate  Superintendent  of  Schools,  Brooklyn 

Arranged  in  Three=Part  or  Two=Part  Series 


The  Three-Part  Series 

Elementary  Arithmetic.        Cloth.  218  pages.        30   cents 

Intermediate  Arithmetic.      Cloth.  252  pages.        35  cents 

Higher  Arithmetic.  Half  leather.    365  pages.        65   cents 

The  Two-Part  Series 

Primary  Arithmetic.  Cloth.  198  pages.    30  cents 

Grammar  School  Arithmetic.     Half  leather.     411  pages.    65  cents 

In  the  Two-Part  Series  the  examination  papers 
are  omitted,  considerably  reducing  the  bulk,  but  in 
no  way  interfering  with  the  completeness  of  the  course. 

The  Walsh   Arithmetics 

INSURE 

Rapid  and  Accurate  Computation 

GIVE 

Constant  Review  without  Repetition 

Abundant  and  Varied  Problems 

OMIT 

Nothing  Essential,  Everything  Else 

ARE 

Fresh,  Well  Graded,  and  Teachable 

EMBODY 

the  Recommendations  of  the  Committee  of  Ten 

AS  WELIv  AS 

of  the  Committee  of  Fifteen 


The   Walsh   Arithmetics 


over  twenty  years  before  his  series  of  arith- 
metics were  published  John  H.  Walsh  had  been 
a  deep  and  philosophic  student,  both  in  and  out  of  the 
classroom,  of  the  faults  of  the  old  system  of  presenting 
arithmetic  and  of  the  features  which  should  characterize 
the  modern  and  effective  text-book. 

How  far  he  had  advanced  on  the  right  road  was 
shown  by  the  fact  that,  when  the  Report  of  the  Com- 
mittee of  Ten  was  issued,  his  arithmetics,  then  in  press, 
were  found  to  have  anticipated  and  practically  embodied 
all  the  important  recommendations  of  that  report. 
Again,  with  the  recommendations  of  the  Committee 
of  Fifteen,  the  agreement  is  quite  as  close. 

These  facts  show  that  sound  principles  underlie 
the  Walsh  Arithmetics.  Further  evidence  is  that 
several  series  of  arithmetics  have  appeared  since,  built 
on  the  same  principles,  but  it  is  worth  noting  that  no 
one  of  these  imitators  has  produced  books  as  faultless 
as  those  of  Mr.  Walsh. 


Features  of  Especial  Merit 


Division   of  Work 

'T^HE  work  of  this  arithmetic  is  divided  into 
sixteen  chapters,  each  containing  a  half-year 
course.  The  elements  of  all  the  important  topics  are 
taken  up  early,  and  treated  more  fully  in  each  succeed- 
ing chapter.  Advanced  work  in  all  the  lines  with 
oral  and  written  drill  is  given  in  each  chapter  with 
special  reviews. 


No   Unnecessary   Rules 

/CHILDREN  learn  the  quick  and  accurate  work- 
ing of  processes  long  before  they  can  compre- 
hend the  underlying  principles,  and  it  is  a  mistake  to 
misuse  valuable  time  upon  the  unnecessary  memorizing 
of  rules  and  definitions.  True  education  will  stimu- 
late the  mental  activity  of  the  child  by  helping 
him  to  work  out  his  own  rules  inductively.  He 
will  then  stand  on  his  feet  securely,  and  what  he 
knows  he  will  know  well.  The  real  educator  should 
be  the  living  teacher ;  the  text-book  but  an  instrument. 

4 


The   Method  of  Grading 

'T'HE  old  method  was  to  take  up  one  topic 
after  another,  beginning  with  addition,  and 
to  exhaust  each  one  before  going  on  to  the  next. 
Under  this  system  many  pupils  who  left  school 
before  the  end  of  the  course  had  no  knowledge  at  all 
of  certain  subjects  which  would  be  of  great  practical 
value  to  them  —  as  percentage,  denominate  numbers, 
etc.  On  the  other  hand,  experience  has  shown  that 
many  graduates  cannot  add,  subtract,  multiply,  and 
divide  with  facility  and  accuracy,  from  too  little  practice 
during  the  later  years  of  school  life. 

Mr.  Walsh  has  taken  what  may  be  called  "  the 
spiral  plan,"  or,  as  the  French  express  it,  the  "concen- 
tric circle  method."  The  book  is  divided,  not  by 
topics  but  by  half-year  courses.  Practice  in  each 
subject  is  carried  on  with  problems  of  gradually 
increasing  difficulty  through  the  whole  course.  Drill 
in  the  four  fundamental  processes  is  found  in  all  the 
chapters.  On  the  other  hand,  certain  subjects,  as 
mensuration  and  denominate  numbers,  are  begun  much 
earlier,  in  simple  form,  of  course,  than  in  other 
books,  so  that  pupils  who  do  not  finish  the  course 
will  have  a  much  better  grasp  of  the  whole  subject 
of  arithmetic  and  be  better  fitted  to  apply  its 
principles  in  the  practical  business  of  life. 


Great  Number  of  Problems 

TNCLUDING  drill  exercises  and  oral  work,  there 
are  over  7600  exercises  and  problems  in  the  first 
book  alone.  The  second  book  contains  a  propor- 
tionate amount  of  practice  work.  No  other 
arithmetic  has  nearly  as  many  exercises  and  practical 
problems.  Teachers  who  use  the  Walsh  Arithmetics 
do  not  need  to  search  for  extra  material  to  give  their 
classes  sufficient  practice. 


Variety  of  Problems 

UACH  problem  is  unlike  the  previous  one,  and 
will  require  the  pupil  to  read  it  carefully. 
He  cannot  work  it  by  referring  to  a  "sample"  at 
the  head  of  the  page.  Variety  is  also  given  by 
the  problems  in  a  large  number  of  examination 
papers  from  many  sources,  which  are  included  in 
the  Three-Book  Course.  Papers  are  given  which 
have  been  used  in  national,  state,  and  municipal 
civil  service  examinations,  as  well  as  in  school  and 
college  examinations. 


Oral  and   Sight  Work 

A  GREAT  amount  of  oral  work  is  given  all 
through  these  books,  both  under  the  several 
topics  discussed  and  in  the  continuous  reviews. 
The  amount  and  variety  of  this  oral  work  covers 
all  the  ground  of  a  mental  arithmetic  and  answers 
all  the  demands  for  a  manual  of  this  subject. 

In  addition  to  the  usual  oral  work,  practice  in 
sight  work  has  been  introduced.  Problems  are  put 
upon  the  blackboard  by  the  teacher,  the  pupils 
perform  the  necessary  processes  mentally  and  write 
the  answers  as  quickly  as  possible.  This  is  a 
helpful  form  of  drill. 

Approximate  Answers 

pRACTICE  in  approximate  answers  will  prove 
to  be  of  the  greatest  practical  value,  for,  in 
business  life,  one  cannot  always  stop  to  reckon 
with  paper  and  pencil,  but  must  be  ready  to 
estimate  quickly  the  approximate  result  in  order 
to  make  a  decision.  By  such  practice  in  rapid 
computation  pupils  learn  to  note  all  conditions  of  a 
problem  and  soon  detect  at  once  the  absurdity  of 
any  result  wide  of  the  mark.  This  useful  drill  is 
found  in  no  other  series  of  arithmetics. 

7 


Continuous   Reviews 

HE  RE  are  not  only  frequent  but  continuous 
reviews,  which  constantly  apply  all  that  has 
been  learned  without  actual  repetition  of  material 
previously  used.  Pupils  thus  cannot  lose  their 
readiness  in  the  application  of  the  simpler  and 
more  fundamental  processes.  Each  half-year's 
work  contains  its  own  review  chapter  and  the 
lessons  arranged  for  this  purpose  will  be  especially 
appreciated  by  teachers. 

Algebra    Work 

JV/FODERN  educational  methods,  as  recommended 
by  the  various  committees  who  have  made 
a  special  study  of  elementary  school  problems,  all 
require  the  introduction  of  simple  algebra  work  in 
the  grammar  grades.  The  algebra  work  included 
in  the  Walsh  Arithmetics,  in  Chapters  X  and  XV, 
is  within  the  capacity  of  the  average  pupil  and 
sufficiently  full  to  give  him  all  the  training  which 
it  is  desirable  for  him  to  have  in  this  grade  of  work. 
It  perfectly  meets  the  recommendations  of  the 
Committee  of  Ten  and  the  Committee  of  Fifteen. 
To  enable  a  teacher  unfamiliar  with  this  work  to 
do  it  successfully,  he  has  only  to  follow  the  lines 
laid  down  in  the  manual. 

8 


Elementary     Geometry 

'pLEMENTARY  Geometry  was  recommended  by 
the  Committee  of  Ten  above  referred  to  and  is 
now  generally  taught  in  the  upper  grammar  grades. 
The  chapter  on  geometry  in  the  Walsh  books  con- 
tains sufficient  material  for  two  years'  work.  Pupils 
who  have  mastered  the  work  presented  in  this  chapter 
will  find  little  difficulty  in  solving  the  questions  in 
inventional  geometry  commonly  offered  in  examina- 
tion papers. 

General    Statement 

TN  the  lower  book  the  alternation  of  oral,  sight,  and 
written  work,  the  early  introduction  of  United 
States  currency  (leading  to  decimals),  the  easy  begin- 
nings with  fractions  and  denominate  numbers,  are 
advantages  which  cannot  be  too  strongly  emphasized. 

In  the  higher  book  the  range  of  subjects,  the  prac- 
tical problems,  the  early  introduction  of  percentage 
and  simple  interest,  of  bills  and  receipts,  and  all  the 
matter  connected  with  simple  commercial  arithmetic, 
together  with  the  short  methods  for  approximate 
answers,  are  notable  points  of  superiority. 

The  series  is  thoroughly  up  to  the  demands  of 
modern  pedagogy.  It  is  inductive  in  method  and 
aims  to  develop  clear  thought  and  the  power  of 
accurate  computation  and  to  lay  the  foundation  for  the 
intelligent  use  of  mathematical  principles. 

9 


Weighty  Words  of  Educators 

ALBERT  LEONARD,   Pres.  Mich.  Nor.  Schools,  Tpsilanti,  Mich. 
Walsh's  Arithmetics  embody   the  best  ideas  of  modern  educational 
philosophy  and  are  a  distinct  improvement  upon  the  older  text-books. 
I  do  not  know  of  any  better  books  for  school  use  than  this  series. 

W.  V.   HAILMAN,   Superintendent,   Dayton,    Ohio. 

The  Walsh  Arithmetics  are  highly  satisfactory  to  me  in  every 
respect.  They  are  eminently  practical  and  free  from  pernicious 
puzzles.  I  appreciate  the  attention  which  the  books  pay  to  the 
place  of  arithmetic  in  mensuration  and  in  industrial  pursuits,  and 
the  effective  manner  in  which,  in  the  advanced  grades,  they  make 
the  transition  to  considerations  of  general  arithmetic  or  algebra. 

C.   W.   CRUIKSHANK,   Superintendent,  Fort  Madison,   Iowa. 

The  books  are  standing  the  test  of  the  classroom  admirably.  I  find 
them  well  graded,  am  well  pleased  with  the  amount  of  work  furnished, 
and  believe  that  the  "  Spiral  Method"  is  the  correct  method.  The 
books  require  independent  work  on  the  part  of  the  pupils.  I  am 
growing  to  like  them  better  as  I  see  the  results  of  their  use  in  our 
schools. 
B.  E.  JACKSON,  Superintendent,  West  Superior,  Wis. 

The  Walsh  Arithmetics  have  been  in  use  in  this  city  for  a  number 
of  years,  and  they  stand  higher  in  the  estimation  of  the  Board, 
teachers,  and  patrons  of  the  school  than  ever  before.  I  consider 
them  the  best  series  published,  and  heartily  endorse  the  "Spiral 
Plan  ' '  as  sound  pedagogical  ly. 
W.  McK.  VANCE,  Superintendent,  Urbana,  Ohio. 

My  teachers  without  exception  are  earnest  admirers  and  hearty 
supporters  of  the  spiral  plan  of  teaching  arithmetic,  as  it  appears 
in  the  Walsh  series.  The  problems  are  graded  with  such  nicety 
that  the  pupil  meets  every  difficulty  with  an  increasing  sense  of 
power.  I  commend  the  books  heartily  and  without  reservation. 

10 


I.   C.   PHILLIPS,   Superintendent,   Lewiston,   Maine. 

We  have  used  the  Walsh  Arithmetics  for  three  years.  Teachers 
and  pupils  are  well  pleased  with  them,  and  the  results  are  better 
than  I  have  ever  obtained  with  any  other  series  of  arithmetics. 

W.   N.   LISTER,    County  Com.  Schools,  Ann  Arbor,  Mich. 

The  Walsh  Arithmetics  are  now  in  very  general  use  throughout 
the  county  and  I  have  yet  to  hear  the  first  unfavorable  criticism 
from  my  teachers.  On  the  contrary,  they  are  expressing  them- 
selves gratified  with  the  results  obtained  from  the  new  plan. 

S.   A.   FARNSWORTH,   Principal,   St.  Paul,   Minn. 

Four  years  ago  the  Walsh  Arithmetics  were  adopted  for  the 
St.  Paul  schools.  When  the  period  of  adoption  expired  last 
summer,  the  principals  of  the  forty-five  schools  of  the  city  were 
practically  unanimous  in  recommending  their  continuance.  They 
are  eminently  adapted  to  foster  careful  and  accurate  reasoning 
along  mathematical  lines.  A  marked  improvement  has  been 
shown  in  the  subject  since  we  began  the  use  of  the  books. 

C.  D.  GRAWN,   Prin.   Tpsilanti  Normal  Training  School. 

During  the  twenty  years  of  my  experience  as  a  teacher  and  a 
superintendent  I  have  never  before  made  use  of  a  series  of  arithmetics 
that  have  enabled  us  to  reach  such  gratifying  results. 

The  books  are  well  graded,  afford  frequent  opportunities  for 
review,  have  well-selected  and  practical  exercises,  and  in  the  second 
book  of  the  grammar  course  give  the  pupil  a  good  working  knowledge 
of  the  algebraic  equation  and  an  intelligent  understanding  of  the 
concepts  of  elementary  geometry,  both  of  which  features  fully 
comply  with  the  recommendations  of  the  Committee  of  Ten. 

F.  S.   SUTCLIFFE,   Superintendent,  Arlington,  Mass. 

I  have  for  two  years  supervised  the  work  of  classes  using 
Walsh  Arithmetics,  and  for  definite,  reliable  results  I  count  them 
the  best  books  I  ever  used. 

•ii 


"  FULLY  A  YEAR  IN  ADVANCE  " 

ELGIN,    ILL. 
D.   C.    HEATH    &    CO.  : 

We  have  been  using  the  Walsh  Arithmetics  in  our 
schools  for  several  years,  and  are  pleased  with  the 
books  because  of  what  they  have  done  in  making  our 
arithmetic  work  more  efficient.  Our  pupils  are  fully 
a  year  in  advance  of  what  they  were  when  the  books 
were  introduced,  —  grade  for  grade,  I  mean. 

Permit  me  to  suggest  a  few  of  the  strong  points  of 
these  books  from  the  standpoint  of  teachers  who  are 
using  them  :  — 

1.  A    large   number    of  simple   problems. 

2.  Frequent    reviews. 

* 

3.  An  abundance  of  oral  work,  thus  doing  away  with  the  neces- 
sity   for    a    separate    book  in   mental  arithmetic. 

4.  Early    introduction    of  easy    work    in   fractions,    denominate 
numbers,    percentage,    and    interest. 

5.  Simple    explanation. 

6.  Distribution    of  elementary  work    in   algebra   and    geometrical 
measurements    and    constructions    so    that    it     may     supplement    and 
elucidate    the    work   in    arithmetic. 

7.  Beginning    algebra    work   with    equation. 

Our  teachers  find  that  this  work  very  greatly  aids 
clear  thinking  and  accuracy  of  statement.  I  unhesi- 
tatingly recommend  the  books. 

M.   A.   WHITNEY,  Supt.   of  Schools. 


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